#8925298

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

3998 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,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, 8832494);