#7030342

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,30,43)(20,25,35,42)(21,27,22,26)(24,36,38,32)(31,45,33,48)(34,44,37,41)(39,46,40,47);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(19,30)(24,38)(29,43)(31,33)(32,36)(39,40)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,36,23,30,22,32)(18,24,26,43,28,38,27,29)(20,33,37,47,35,31,34,46)(25,40,44,45,42,39,41,48);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20,23,35)(18,25,28,42)(19,31,30,33)(21,37,22,34)(24,39,38,40)(26,44,27,41)(29,45,43,48)(32,47,36,46);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,32,30,36)(20,34,35,37)(24,29,38,43)(25,41,42,44)(31,46,33,47)(39,48,40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,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, 6937538);