#124697

order	512
nilpotency class	4
isoclinism	isoclinic class of #124657
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵ ◁ C₂²×#95 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 646 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)(19,25)(20,28,30,23)(21,24)(22,27,31,29)(26,32);(2,5)(3,8)(6,7)(17,19,21,26)(18,22)(20,29,30,27)(23,32)(24,31)(25,28);(10,13)(14,16)(19,26)(22,31)(25,32)(27,29)

Polycyclic presentation:

2^9|a^2=e,b^2=fg,d^2=gi,e^2=h,b^a=bd,c^a=ch,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efgi,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, 31893);