#495908

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	C₂³	C₂³×C₈
prime radical	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=g,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, 403104);