#391199

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1258 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)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,28)(24,26)(25,35)(27,29)(30,33)(31,36)(32,34);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,23,26,32)(22,28,24,34)(25,33,27,31)(29,36,35,30);(5,7)(6,10)(8,12)(9,11)(14,18)(17,20)(21,24,26,22)(23,30,32,36)(25,35,27,29)(28,31,34,33);(7,12)(9,10)(15,19)(17,20)(21,25,26,27)(22,29,24,35)(23,31,32,33)(28,30,34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=def,g^b=gi,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, 298395);