#10521293

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

4327 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,14)(15,16)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28)(33,34,40,43)(35,44,38,41)(36,39,37,42)(45,46,47,48);(2,7)(3,11)(9,12)(10,16)(33,35,40,38)(34,41,43,44)(36,47,37,45)(39,46,42,48);(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30)(33,36,40,37)(34,42,43,39)(35,45,38,47)(41,46,44,48);(1,3,6,11)(2,8,7,13)(4,10,14,16)(5,12,15,9)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28)(33,37,40,36)(34,39,43,42)(35,45,38,47)(41,46,44,48);(1,4)(2,9)(3,10)(5,8)(6,14)(7,12)(11,16)(13,15)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30)(33,38,40,35)(34,41,43,44)(36,47,37,45)(39,48,42,46);(1,5)(2,10)(3,12)(4,13)(6,15)(7,16)(8,14)(9,11)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)(33,39)(34,36)(35,46)(37,43)(38,48)(40,42)(41,47)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=ghi,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428489);