#8110774

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

4372 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017970);