#7308720

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#24124	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1473 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^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=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7215916);