#7312358

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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	#42673	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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)(3,6)(4,5)(7,8)(17,18,23,25,22,27,21,28)(19,24,34,36,30,39,31,38)(20,26,35,40,32,42,33,41)(29,37,45,46,44,47,43,48);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,34)(25,36)(26,37)(27,39)(28,38)(32,44)(33,43)(35,45)(40,46)(41,48)(42,47);(9,11)(10,13)(12,16)(14,15)(18,25)(21,23)(24,36)(26,40)(27,28)(31,34)(33,35)(37,46)(38,39)(41,42)(43,45)(47,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,32)(18,26,27,42)(19,29,30,44)(21,33,23,35)(24,37,39,47)(25,40,28,41)(31,43,34,45)(36,46,38,48);(2,5)(6,8)(10,14)(13,15)(18,27)(19,30)(20,32)(25,28)(31,34)(33,35)(37,47)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7219554);