#386395

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=efgi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293591);