#7328140

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#45196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1435 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7235336);