#6473899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6381095);