#225749

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂³	D₄×Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

531 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)(4,12)(7,13)(8,15)(9,16)(10,14)(17,18,21,24,22,25,20,30)(19,26,28,31,29,32,27,23);(1,2,5,7)(3,9,11,15)(4,13,12,6)(8,14,16,10);(2,7)(6,13)(8,16)(9,15)(17,19,20,27,22,29,21,28)(18,23,24,26,25,31,30,32);(1,3,5,11)(2,8,7,16)(4,14,12,10)(6,15,13,9)

Polycyclic presentation:

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

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, 132945);