#8821775

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

4124 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=h,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, 8728971);