#124737

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

320 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31933);