#124640

order	512
nilpotency class	4
isoclinism	isoclinic class of #11836
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1514
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂	#7604
prime radical	C₂³	#421
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

247 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,12,15,24,16,30,22,32)(4,17,18,29)(5,19,21,9)(6,11)(7,10)(8,14,25,31,26,13,27,20)(23,28);(1,3)(2,8,10,26)(4,13)(5,20)(6,22)(7,16)(9,24,19,32)(11,25,28,27)(12,29,30,17)(14,18)(15,23)(21,31);(1,4,6,5,7,18,23,21)(2,9,28,29,10,19,11,17)(3,13,22,20,16,14,15,31)(8,24,27,30,26,32,25,12)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fgh,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,d^c=di,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, 31836);