#7325878

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	#53064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=i,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, 7233074);