#356941

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

1078 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)(21,22,25,29,26,30,34,36)(23,27,24,32,31,35,33,28);(1,2,3,4)(6,9)(7,11)(13,15)(14,17)(16,20)(18,19)(23,31)(24,33)(27,35)(28,32);(14,18)(17,19)(21,23,25,24,26,31,34,33)(22,27,29,32,30,35,36,28);(5,7,8,11)(6,10,9,12)(15,20)(17,19)(21,24,25,31,26,33,34,23)(22,28,29,27,30,32,36,35)

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^a=di,c^b=ci,d^b=dei,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264137);