#7160719

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

1376 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7067915);