#7536308

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1619 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)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,26)(21,27)(22,28)(23,29)(24,32)(30,41)(31,42)(33,48)(34,39)(35,44)(36,45)(37,46)(38,40)(43,47);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,33)(21,31)(22,32)(23,38)(25,28)(26,43)(27,40)(29,42)(30,37)(34,35)(36,47)(39,46)(41,44)(45,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,25)(20,34)(21,38)(22,32)(23,31)(24,28)(26,39)(27,40)(29,42)(30,36)(33,35)(37,47)(41,45)(43,46)(44,48);(10,14)(13,15)(17,20,22,36)(18,26,28,45)(19,30,32,34)(21,37,23,35)(24,39,25,41)(27,46,29,44)(31,47,38,33)(40,48,42,43);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,31,32,38)(20,35,36,37)(24,40,25,42)(26,44,45,46)(30,33,34,47)(39,43,41,48)

Polycyclic presentation:

2^9|a^2=g,c^2=fgi,d^2=i,e^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, 7443504);