#215748

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122944);