#8097795

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

4286 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004991);