#124941

order	512
nilpotency class	4
isoclinism	isoclinic class of #124940
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

464 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 90 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)(4,7)(6,8)(9,10,14,21,16,24,40,20,17,22,39,52,15,23,13,38)(11,25,28,45,30,56,27,46,31,55,63,18,29,44,41,47)(12,32,34,53,36,50,33,48,37,19,42,49,35,51,67,66)(26,57,59,65,61,72,68,70,62,54,64,43,60,69,58,71);(2,5)(3,8)(6,7)(9,11,17,31)(10,18,22,45)(12,26,37,62)(13,28,40,63)(14,41,39,27)(15,30,16,29)(19,43,32,65)(20,44,38,56)(21,25,52,55)(23,47,24,46)(33,64,67,59)(34,58,42,68)(35,61,36,60)(48,72,66,69)(49,57,53,54)(50,71,51,70);(9,12)(10,19)(11,26)(13,33)(14,42)(15,35)(16,36)(17,37)(18,43)(20,48)(21,53)(22,32)(23,50)(24,51)(25,54)(27,58)(28,64)(29,60)(30,61)(31,62)(34,39)(38,66)(40,67)(41,68)(44,72)(45,65)(46,70)(47,71)(49,52)(55,57)(56,69)(59,63)

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=gi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32137);