#224560

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

887 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)(9,10,13,16,14,11,12,15)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,30)(31,32);(1,2,5,6)(3,7,4,8)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,31,25,30);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,16,13,15)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(1,4,5,3)(2,7,6,8)(9,12,14,13)(10,15,11,16)(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131756);