#416338

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

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, 323534);