#7532105

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	#36011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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)(17,18,22,27)(19,29,32,24)(20,25,21,26)(23,28,37,45)(30,43,35,39)(31,47,48,40)(33,46,38,41)(34,42,36,44);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,32)(18,24,27,29)(20,30,21,35)(23,38,37,33)(25,39,26,43)(28,46,45,41)(31,34,48,36)(40,42,47,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,25)(19,30)(21,22)(23,34)(24,39)(26,27)(28,42)(29,43)(31,33)(32,35)(36,37)(38,48)(40,41)(44,45)(46,47);(10,14)(13,15)(17,21,22,20)(18,26,27,25)(19,31,32,48)(23,34,37,36)(24,40,29,47)(28,42,45,44)(30,38,35,33)(39,46,43,41);(2,5)(6,8)(10,14)(13,15)(20,21)(25,26)(30,35)(31,48)(34,36)(39,43)(40,47)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=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, 7439301);