#375616

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

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

1152 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)(7,11)(8,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,26,24,25,28,33,32)(23,27,31,29,30,34,36,35);(6,9)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(2,4)(15,20)(17,19)(21,24,26,28,25,32,33,22)(23,29,31,34,30,35,36,27);(5,7,8,12)(6,10,9,11)(14,18)(17,19)(23,30)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di

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, 282812);