#230493

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 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:

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

Polycyclic presentation:

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

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, 137689);