#356929

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=efg,g^2=i,b^a=be,c^a=cf,d^b=dei,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, 264125);