#7584740

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1523 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:

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

Polycyclic presentation:

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

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, 7491936);