#10457912

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

4018 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)(11,12)(13,15)(14,16)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(33,34)(35,40)(36,38)(37,44)(39,43)(41,45)(42,46)(47,48);(2,7)(3,10)(9,15)(11,16)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,f^2=gh,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, 10365108);