#7046521

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6953717);