#7224219

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#53074	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1646 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=fh,e^2=fi,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, 7131415);