#8455997

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

4199 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8363193);