#362272

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gi

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, 269468);