#7090233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#24438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,24,28)(19,29,34,47)(20,35,21,26)(22,27,23,39)(25,32,42,33)(30,41,37,45)(31,46,48,40)(36,44,38,43);(9,10)(11,15)(12,14)(13,16)(17,19,22,32,24,34,23,33)(18,25,39,47,28,42,27,29)(20,30,38,48,21,37,36,31)(26,40,43,41,35,46,44,45);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,21)(18,26,28,35)(19,30,34,37)(22,38,23,36)(25,40,42,46)(27,44,39,43)(29,45,47,41)(31,32,48,33);(11,16)(13,15)(17,21,24,20)(18,26,28,35)(19,31,34,48)(22,38,23,36)(25,41,42,45)(27,43,39,44)(29,46,47,40)(30,32,37,33);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,27,28,39)(19,32,34,33)(20,36,21,38)(25,29,42,47)(26,43,35,44)(30,31,37,48)(40,41,46,45)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6997429);