#7328202

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324564
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	#45101	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1441 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)(19,27)(20,25)(21,23)(22,35)(24,31)(26,29)(28,42)(30,41)(32,46)(33,37)(34,48)(36,43)(38,45)(39,47)(40,44);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(18,23)(19,28)(25,37)(26,35)(30,32)(31,43)(40,48)(45,47);(9,11)(10,13)(12,16)(14,15)(17,19,29,43,21,28,22,31)(18,24,35,42,23,36,26,27)(20,32,44,45,33,30,34,47)(25,39,48,41,37,38,40,46);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,30)(21,33)(22,34)(23,37)(24,38)(26,40)(27,41)(28,32)(29,44)(31,45)(35,48)(36,39)(42,46)(43,47);(2,5)(6,8)(10,14)(13,15)(18,23)(20,33)(24,36)(26,35)(27,42)(30,32)(34,44)(45,47)

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,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, 7235398);