#8097821

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

4134 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)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,48)(46,47);(1,2)(3,8)(4,7)(5,9)(6,11)(10,15)(12,14)(13,16)(20,29)(24,28)(26,32)(30,31)(33,35)(34,40)(36,46)(37,43)(38,47)(39,48)(41,42)(44,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,39,43)(34,41,44,38)(35,46,48,37)(40,42,45,47);(1,4)(2,7)(3,10)(5,12)(6,13)(8,15)(9,14)(11,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,16)(13,14)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,37,39,46)(34,42,44,47)(35,43,48,36)(38,40,41,45);(1,2)(3,6)(4,7)(5,9)(8,11)(10,13)(12,14)(15,16)(17,19)(18,25)(20,26)(21,23)(22,27)(24,31)(28,30)(29,32)(33,38,39,41)(34,43,44,36)(35,47,48,42)(37,45,46,40)

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=gh,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, 8005017);