#7531685

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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 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)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,22)(19,24,29,35)(20,25,31,32)(21,26,34,36)(27,37,41,45)(28,38,42,33)(30,39,44,46)(40,47,48,43);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,22,35)(20,27,31,41)(21,33,34,38)(25,37,32,45)(26,28,36,42)(30,43,44,47)(39,40,46,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,27)(21,30)(22,32)(23,31)(24,37)(26,39)(28,40)(29,41)(33,43)(34,44)(35,45)(36,46)(38,47)(42,48);(10,14)(13,15)(17,21,23,34)(18,26,22,36)(19,28,29,42)(20,30,31,44)(24,38,35,33)(25,39,32,46)(27,40,41,48)(37,47,45,43);(2,5)(6,8)(10,14)(13,15)(20,31)(25,32)(27,41)(30,44)(37,45)(39,46)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=i,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,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, 7438881);