#8425335

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

4278 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8332531);