#6474984

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fg,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6382180);