#6543180

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6450376);