#124121

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#95	C₂³
socle	C₂³	C₂×#99
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

634 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31317);