#144371

order	512
nilpotency class	4
isoclinism	isoclinic class of #144369
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

857 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 857 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,6)(7,8)(9,10)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,51)(24,49)(25,50)(26,57)(27,39)(28,40)(29,41)(30,43)(31,44)(32,64)(33,46)(34,47)(35,52)(36,54)(37,55)(38,58)(42,59)(45,62)(48,53)(56,68)(60,67)(61,70)(63,69)(65,71)(66,72);(5,8)(9,11,15,24)(10,16,20,37)(12,26,29,56)(13,23,30,55)(14,32,34,25)(17,38,41,66)(18,21,43,49)(19,45,47,36)(22,33,53,28)(27,58,59,72)(31,62,61,54)(35,46,65,40)(39,57,42,68)(44,64,70,50)(48,67,51,69)(52,63,71,60);(1,3)(2,5)(4,7)(6,8)(9,12,15,29)(10,17,20,41)(11,22,24,53)(13,27,30,59)(14,33,34,28)(16,35,37,65)(18,42,43,39)(19,46,47,40)(21,48,49,51)(23,52,55,71)(25,56,32,26)(31,63,61,60)(36,66,45,38)(44,67,70,69)(50,57,64,68)(54,72,62,58)

Polycyclic presentation:

2^9|b^2=i,c^2=i,f^2=i,b^a=bd,c^a=ce,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,d^c=dg,f^c=fi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209847
maximal normal subgroup of	#72209847
maximal quotient of	#72227686, #72227720, #72227769, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51567);