#180956

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1339 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=i,b^a=be,e^a=eh,g^a=gh,c^b=ch,d^c=dhi

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, 88152);