#8425525

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

4174 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(18,23)(24,25)(26,29)(31,32)(33,34,38,39)(35,42,37,40)(36,44,46,41)(43,47,45,48);(2,7)(3,10)(8,14)(11,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(34,39)(40,42)(41,44)(47,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21,22,30)(18,26,23,29)(19,27,20,28)(24,31,25,32)(33,37,38,35)(34,40,39,42)(36,45,46,43)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=gh,f^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, 8332721);