#498633

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

514 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)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27,23,28,38,44,21,26,29,39,37,19,24,31)(20,32,35,46,36,30,42,45,34,25,41,43,48,33,40,47);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,19,29,44,23,18,24,39,21,28,22,31,37,26,38,27)(20,33,41,45,36,32,40,43,34,30,35,47,48,25,42,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,34)(18,25,26,32)(19,30,28,33)(22,35,29,41)(23,36,37,48)(24,40,38,42)(27,43,39,46)(31,45,44,47);(2,5)(6,8)(11,16)(13,15)(20,34)(25,32)(30,33)(35,41)(36,48)(40,42)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,d^b=de,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, 405829);