#124217

order	512
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂³	C₂×#102
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

430 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31413);