#466403

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=fg,c^2=ei,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373599);