#8092714

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999910);