#124311

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 680 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31507);