#7038827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42760	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=f,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=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6946023);