#356049

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 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)(6,9)(7,11)(13,14)(15,17)(16,18)(19,20)(21,22,25,30,26,29,27,24)(23,31,35,36,34,28,33,32);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,17,18,20)(21,23)(22,28)(24,36)(25,35)(26,34)(27,33)(29,31)(30,32);(6,9)(10,12)(14,18)(17,20)(21,24,27,29,26,30,25,22)(23,32,33,28,34,36,35,31);(5,7,8,11)(6,10,9,12)(21,25,26,27)(22,24,29,30)(23,33,34,35)(28,32,31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=h,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=ce,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263245);