#8821781

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

4198 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)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18,22,23)(19,21,26,24)(20,25,28,30)(27,29,32,31)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(2,6)(7,9)(8,13)(14,15)(18,23)(20,28)(21,24)(27,32)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(3,10)(4,12)(7,9)(8,13)(18,23)(19,26)(20,28)(29,31)(34,35)(36,43)(40,42)(44,48);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,32)(29,30)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32)(33,36)(34,39)(35,41)(37,44)(38,43)(40,46)(42,47)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,22,24)(18,19,23,26)(20,29,28,31)(25,27,30,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=gh,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8728977);