#122835

order	512
nilpotency class	4
isoclinism	isoclinic class of #122714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

277 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=g,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30031);