#7532069

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 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)(17,18,22,23)(19,27,30,24)(20,25,32,33)(21,26,36,37)(28,42,44,38)(29,34,46,39)(31,40,35,41)(43,47,45,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,23,27)(20,28,32,44)(21,34,36,39)(25,38,33,42)(26,29,37,46)(31,47,35,48)(40,43,41,45);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,32)(18,25,23,33)(19,28,30,44)(21,35,36,31)(24,38,27,42)(26,41,37,40)(29,45,46,43)(34,48,39,47);(10,14)(13,15)(17,21,22,36)(18,26,23,37)(19,29,30,46)(20,31,32,35)(24,39,27,34)(25,40,33,41)(28,43,44,45)(38,48,42,47);(2,5)(6,8)(10,14)(13,15)(20,32)(25,33)(28,44)(31,35)(38,42)(40,41)(43,45)(47,48)

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=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7439265);