#378704

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×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

976 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=ef,d^2=efh,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=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, 285900);