#7581911

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,33)(22,26)(23,27)(25,35)(28,46)(29,40)(31,39)(32,44)(34,48)(36,45)(37,43)(38,42)(41,47);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,20)(18,24,26,30)(21,34,35,47)(23,32,37,29)(25,41,33,48)(27,44,43,40)(28,38,31,36)(39,45,46,42);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,22,19)(18,24,26,30)(21,31,35,28)(23,32,37,29)(25,39,33,46)(27,40,43,44)(34,36,47,38)(41,45,48,42);(10,14)(13,15)(17,21,22,35)(18,25,26,33)(19,28,20,31)(23,38,37,36)(24,39,30,46)(27,45,43,42)(29,47,32,34)(40,48,44,41);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,20)(24,30)(28,31)(29,32)(34,47)(39,46)(40,44)(41,48)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7489107);