#362529

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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)(13,14)(15,19)(16,17)(18,20)(21,22)(23,31)(24,28)(25,34)(26,30)(27,29)(32,35)(33,36);(5,7,8,12)(6,9,10,11)(14,17)(18,19)(21,23,25,33,26,24,27,32)(22,28,34,35,30,31,29,36);(14,17)(18,19)(21,24,26,23)(22,28,30,31)(25,33,27,32)(29,35,34,36);(6,10)(9,11)(13,15,16,20)(14,18,17,19)(21,25,26,27)(22,29,30,34)(23,32,24,33)(28,35,31,36)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=i,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=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, 269725);