#472492

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,10,9)(7,12,8,11)(13,14)(15,17)(16,18)(19,20)(21,22,25,28)(23,29,31,26)(24,27,33,34)(30,32,36,35);(5,7,10,8)(6,11,9,12)(13,15,16,19)(14,17,18,20)(21,23,22,26,25,31,28,29)(24,32,34,36,33,35,27,30);(1,2,3,4)(5,8,10,7)(6,11,9,12)(21,24,25,33)(22,27,28,34)(23,30,31,36)(26,35,29,32);(5,9,10,6)(7,12,8,11)(14,18)(17,20)(24,33)(27,34)(30,36)(32,35)

Polycyclic presentation:

2^9|a^2=e,b^2=eghi,c^2=ef,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,g^c=gi

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, 379688);