#124656

order	512
nilpotency class	4
isoclinism	isoclinic class of #124654
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 776 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) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31852);