#7582705

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1279 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)(11,16)(12,14)(13,15)(17,18,23,28)(19,24,32,39)(20,25,34,41)(21,26,22,27)(29,36,45,48)(30,37,31,38)(33,40,35,42)(43,46,44,47);(2,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19,23,32)(18,24,28,39)(20,29,34,45)(21,31,22,30)(25,36,41,48)(26,38,27,37)(33,43,35,44)(40,46,42,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,20,23,34)(18,25,28,41)(19,29,32,45)(21,35,22,33)(24,36,39,48)(26,42,27,40)(30,43,31,44)(37,46,38,47);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(19,30)(21,22)(24,37)(26,27)(29,43)(31,32)(33,35)(36,46)(38,39)(40,42)(44,45)(47,48);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,21,23,22)(18,26,28,27)(19,31,32,30)(20,33,34,35)(24,38,39,37)(25,40,41,42)(29,43,45,44)(36,46,48,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7489901);