#8925098

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

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

4173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832294);