#7204155

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#36019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18,22,27)(19,29,32,24)(20,26,36,44)(21,28,23,25)(30,42,34,39)(31,38,33,40)(35,41,37,43)(45,48,46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,32)(18,24,27,29)(20,34,36,30)(21,31,23,33)(25,38,28,40)(26,42,44,39)(35,46,37,45)(41,48,43,47);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(18,25)(21,23)(24,38)(26,41)(27,28)(29,40)(31,33)(35,37)(39,47)(42,48)(43,44)(45,46);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,20,22,36)(18,26,27,44)(19,30,32,34)(21,37,23,35)(24,39,29,42)(25,41,28,43)(31,46,33,45)(38,47,40,48);(3,7)(5,8)(10,15)(14,16)(17,21,22,23)(18,25,27,28)(19,31,32,33)(20,35,36,37)(24,38,29,40)(26,43,44,41)(30,45,34,46)(39,48,42,47)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,e^c=ei,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, 7111351);