#7188459

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1228 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)(4,8)(5,7)(10,14)(12,15)(17,18,22,26)(19,27,31,37)(20,24,32,39)(21,34,35,25)(23,29,38,30)(28,41,43,47)(33,46,45,40)(36,42,48,44);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,12,16,15)(17,19,22,31)(18,23,26,38)(20,28,32,43)(21,29,35,30)(24,36,39,48)(25,37,34,27)(33,42,45,44)(40,47,46,41);(2,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,32)(18,24,26,39)(19,28,31,43)(21,33,35,45)(23,36,38,48)(25,40,34,46)(27,41,37,47)(29,42,30,44);(1,4,5,3)(2,8,7,6)(10,14)(11,16)(19,29)(21,35)(23,37)(25,34)(27,38)(28,42)(30,31)(33,45)(36,47)(40,46)(41,48)(43,44);(9,12,13,15)(10,11,14,16)(20,32)(24,39)(28,43)(33,45)(36,48)(40,46)(41,47)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7095655);