#7081782

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1550 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,32,43)(20,33,35,25)(21,27,22,26)(24,37,40,31)(30,46,34,45)(36,42,38,44)(39,48,41,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,37,23,32,22,31)(18,24,26,43,28,40,27,29)(20,34,38,48,35,30,36,47)(25,41,44,45,33,39,42,46);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,30)(21,38)(22,36)(23,35)(24,39)(26,44)(27,42)(28,33)(29,45)(31,48)(32,34)(37,47)(40,41)(43,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,31)(20,35)(21,22)(24,29)(25,33)(26,27)(30,47)(32,37)(34,48)(39,46)(40,43)(41,45);(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,36,35,38)(24,29,40,43)(25,42,33,44)(30,47,34,48)(39,46,41,45)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6988978);