#7230621

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7137817);