#8425336

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

4234 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332532);