#225508

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

557 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=chi,d^b=de,e^b=eh,d^c=di,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, 132704);