#6473745

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1306 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,31,39)(20,25,32,41)(21,26,22,37)(24,35,40,30)(29,45,33,48)(34,43,36,44)(38,46,42,47);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(20,32)(25,41)(29,33)(34,36)(38,42)(43,44)(45,48)(46,47);(11,15)(13,16)(17,19,21,35,23,31,22,30)(18,24,37,39,27,40,26,28)(20,33,36,47,32,29,34,46)(25,42,43,45,41,38,44,48);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,32)(24,38)(26,44)(27,41)(28,45)(30,47)(31,33)(35,46)(37,43)(39,48)(40,42);(10,14)(13,16)(17,21,23,22)(18,26,27,37)(19,30,31,35)(20,34,32,36)(24,39,40,28)(25,43,41,44)(29,46,33,47)(38,45,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380941);