#125171

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
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

386 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,10,14,21,16,24,40,20,17,22,39,53,15,23,13,38)(11,25,28,44,30,56,27,45,31,55,61,18,29,43,41,46)(12,19,34,49,36,52,66,48,37,50,65,71,35,51,33,64)(26,54,58,69,32,62,57,70,60,72,63,42,59,68,67,47);(2,5)(3,8)(6,7)(9,11,17,31)(10,18,22,44)(12,32,37,59)(13,28,40,61)(14,41,39,27)(15,30,16,29)(19,47,50,70)(20,43,38,56)(21,25,53,55)(23,46,24,45)(26,35,60,36)(33,57,66,67)(34,58,65,63)(42,51,69,52)(48,54,64,72)(49,62,71,68);(9,12,15,35,17,37,16,36)(10,19,23,51,22,50,24,52)(11,26,29,59,31,60,30,32)(13,33,39,65,40,66,14,34)(18,42,45,70,44,69,46,47)(20,48,21,49,38,64,53,71)(25,54,43,68,55,72,56,62)(27,57,28,58,41,67,61,63)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32367);