#363989

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,h^c=hi

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, 271185);