#498744

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,28,23,24,36,38,21,27,35,43,19,26,30,42)(20,31,34,45,32,40,46,48,33,25,39,41,29,37,44,47);(1,2,4,5)(3,8,7,6)(9,11)(10,13)(12,15)(14,16)(17,19,21,23)(18,24,27,26)(20,32,33,29)(22,30,35,36)(25,40,31,37)(28,38,43,42)(34,46,39,44)(41,48,45,47);(2,5)(6,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,37)(26,40)(27,31)(28,41)(30,44)(35,39)(36,46)(38,47)(42,48)(43,45);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,19,21,23)(18,26,27,24)(20,32,33,29)(22,30,35,36)(25,37,31,40)(28,42,43,38)(34,46,39,44)(41,47,45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cefi,d^b=dei,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405940);