#7198417

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=fgh,e^2=h,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=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105613);