#10451591

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

3953 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)(3,11)(4,13)(5,10)(6,7)(8,12)(9,14)(15,16)(17,18,21,25)(19,23,27,22)(20,28,29,24)(26,30,32,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(4,14)(8,11)(10,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,38,36)(35,39,43,41)(37,40,45,44)(42,46,48,47);(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3)(2,8)(4,15)(5,13)(6,12)(7,11)(9,16)(10,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,4)(2,9)(3,10)(5,8)(6,14)(7,13)(11,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,5)(2,10)(3,9)(4,8)(6,16)(7,15)(11,14)(12,13)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358787);