#8695952

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

4027 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603148);