#498686

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
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

453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=ef,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=de,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, 405882);