#7081625

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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,30,38)(20,32,33,24)(21,35,36,25)(23,29,39,31)(28,42,45,41)(34,40,48,47)(37,44,46,43);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,31,22,30,36,29)(18,23,25,27,26,39,35,38)(20,28,34,44,33,45,48,43)(24,37,40,41,32,46,47,42);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,40)(26,32)(27,41)(29,43)(30,45)(31,44)(35,47)(36,48)(38,42)(39,46);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,29)(21,36)(23,38)(25,35)(27,39)(28,43)(30,31)(34,48)(37,42)(40,47)(41,46)(44,45);(10,14)(13,15)(19,30)(20,33)(23,39)(24,32)(27,38)(29,31)(34,48)(40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=h,d^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,f^d=fi

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, 6988821);