#497039

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

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

562 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=ce,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, 404235);