#7308529

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1440 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)(17,18,23,28)(19,24,31,42)(20,26,34,44)(21,27,22,37)(25,36,45,33)(29,39,32,46)(30,40,35,41)(38,47,43,48);(1,2)(3,7)(4,6)(5,8)(9,11,14,15)(10,13,16,12)(17,19,23,31)(18,24,28,42)(20,32,34,29)(21,35,22,30)(25,43,45,38)(26,46,44,39)(27,41,37,40)(33,47,36,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20,23,34)(18,25,28,45)(19,29,31,32)(21,33,22,36)(24,38,42,43)(26,27,44,37)(30,47,35,48)(39,41,46,40);(2,6)(3,8)(9,12,14,13)(10,15,16,11)(17,20)(18,26)(19,29)(21,36)(22,33)(23,34)(24,39)(25,37)(27,45)(28,44)(30,48)(31,32)(35,47)(38,40)(41,43)(42,46);(2,6)(5,7)(9,13,14,12)(10,15,16,11)(17,21,23,22)(18,27,28,37)(19,30,31,35)(20,33,34,36)(24,40,42,41)(25,44,45,26)(29,47,32,48)(38,39,43,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=f,e^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=eh,d^c=dh,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, 7215725);