#496140

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator, socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,25,20,22,34,39,18,24,32,40,16,17,23,33)(15,26,30,42,27,28,41,43,29,21,35,37,31,38,36,44);(5,7,9,8)(6,10,11,12)(13,15,18,29)(14,21,24,26)(16,31,20,27)(17,28,22,38)(19,30,32,35)(23,36,34,41)(25,37,40,42)(33,43,39,44);(1,2,3,4)(13,16,18,20)(14,22,24,17)(15,27,29,31)(19,23,32,34)(21,28,26,38)(25,39,40,33)(30,41,35,36)(37,43,42,44);(5,8,9,7)(6,10,11,12)(13,17,32,39,20,14,23,40,18,22,19,33,16,24,34,25)(15,28,35,44,27,21,36,42,29,38,30,43,31,26,41,37)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,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, 403336);