#8097820

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

4298 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)(17,18,23,26)(19,27,21,24)(20,25,31,22)(28,29,30,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(17,18)(19,27)(20,22)(21,24)(23,26)(25,31)(28,32)(29,30)(36,45)(40,44)(42,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,21)(18,24,26,27)(20,28,31,30)(22,29,25,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,20)(18,25)(19,28)(21,30)(22,26)(23,31)(24,32)(27,29)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,23,19)(18,24,26,27)(20,30,31,28)(22,29,25,32)(33,37,38,34)(35,41,43,39)(36,44,45,40)(42,47,48,46);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,22,23,25)(18,20,26,31)(19,29,21,32)(24,28,27,30)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=h,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, 8005016);