#7308721

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

1442 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)(19,29)(20,33)(21,26)(22,27)(23,38)(24,31)(25,34)(28,37)(30,43)(32,41)(35,40)(36,42)(39,48)(44,46)(45,47);(1,2)(3,7)(4,6)(5,8)(9,11,14,15)(10,13,16,12)(17,19)(18,24)(20,30)(21,35)(22,31)(23,32)(25,39)(26,43)(27,29)(28,41)(33,40)(34,46)(36,48)(37,47)(38,45)(42,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,20,22,21)(18,25,27,42)(19,30,31,35)(23,36,37,34)(24,39,29,44)(26,38,33,28)(32,48,47,46)(40,41,43,45);(2,6)(3,8)(9,12,14,13)(10,15,16,11)(17,21,22,20)(18,26,27,33)(19,30,31,35)(23,36,37,34)(24,40,29,43)(25,38,42,28)(32,46,47,48)(39,41,44,45);(2,6)(5,7)(9,13,14,12)(10,15,16,11)(18,27)(19,31)(20,21)(28,38)(32,47)(34,36)(39,44)(40,43)

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=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, 7215917);