#152964

order	512
nilpotency class	5
isoclinism	isoclinic class of #149931
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊Q₈	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ C₂×C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=eh,d^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fgh,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60160);