#8492809

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

4274 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)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,23,26)(19,27,30,24)(20,31,32,21)(22,25,29,28)(33,34,39,43)(35,40,44,48)(36,38,42,47)(37,46,45,41);(3,10)(4,12)(6,7)(8,15)(17,19)(18,24)(20,28)(21,22)(23,30)(25,32)(26,27)(29,31)(35,44)(37,45)(40,48)(41,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,20)(18,21)(19,28)(22,24)(23,32)(25,30)(26,31)(27,29)(33,35,39,44)(34,40,43,48)(36,45,42,37)(38,41,47,46);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20)(18,21)(19,25)(22,27)(23,32)(24,29)(26,31)(28,30)(33,36)(34,38)(35,45)(37,44)(39,42)(40,41)(43,47)(46,48);(1,4,5,12)(2,8,9,15)(3,11,10,16)(6,13,7,14)(17,21)(18,20)(19,29)(22,30)(23,31)(24,25)(26,32)(27,28)(33,37,39,45)(34,41,43,46)(35,36,44,42)(38,40,47,48);(1,2)(3,6)(4,8)(5,9)(7,10)(11,13)(12,15)(14,16)(17,22,23,29)(18,25,26,28)(19,21,30,31)(20,24,32,27)(33,38)(34,42)(35,41)(36,43)(37,40)(39,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8400005);