#7303437

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,33)(21,27)(22,37)(23,28)(25,34)(26,35)(29,43)(30,40)(31,47)(32,41)(36,42)(38,44)(39,45)(46,48);(1,2)(3,7)(4,6)(5,8)(9,11,13,10)(12,15,16,14)(17,19,23,32)(18,24,28,41)(20,29,35,45)(21,30,22,31)(25,38,42,48)(26,39,33,43)(27,40,37,47)(34,44,36,46);(1,3)(2,5)(4,8)(6,7)(10,11)(12,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,38)(26,27)(28,42)(30,44)(31,46)(32,45)(33,37)(39,40)(41,48)(43,47);(2,6)(3,8)(9,12,13,16)(10,14,11,15)(17,20)(18,26)(19,29)(21,36)(22,34)(23,35)(24,39)(25,27)(28,33)(30,46)(31,44)(32,45)(37,42)(38,40)(41,43)(47,48);(2,6)(5,7)(10,11)(14,15)(17,21,23,22)(18,27,28,37)(19,30,32,31)(20,34,35,36)(24,40,41,47)(25,26,42,33)(29,44,45,46)(38,39,48,43)

Polycyclic presentation:

2^9|b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 7210633);