#8919488

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

4249 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34,38,41)(35,40,44,39)(36,45,46,37)(42,43,47,48);(2,7)(4,8)(9,13)(12,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,48)(46,47);(2,7)(3,10)(5,14)(12,15)(18,19)(21,29)(23,25)(27,32)(36,46)(37,45)(42,47)(43,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,36)(34,37)(35,42)(38,46)(39,43)(40,48)(41,45)(44,47);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,35)(34,40)(36,47)(37,43)(38,44)(39,41)(42,46)(45,48);(1,5,6,14)(2,9,7,13)(3,11,10,16)(4,12,8,15)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,36)(35,43)(38,45)(39,42)(40,47)(41,46)(44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826684);