#10480957

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

4212 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)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31)(33,34,39,43)(35,41,42,38)(36,45,37,40)(44,46,47,48);(2,7)(3,10)(9,15)(11,16)(19,20)(23,24)(27,28)(31,32)(33,35)(34,38)(36,44)(37,47)(39,42)(40,46)(41,43)(45,48);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36,39,37)(34,40,43,45)(35,44,42,47)(38,46,41,48);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,12,15,14)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,37,39,36)(34,40,43,45)(35,44,42,47)(38,48,41,46);(1,4)(2,9)(3,11)(5,14)(6,13)(7,15)(8,12)(10,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,35)(34,41)(36,44)(37,47)(38,43)(39,42)(40,48)(45,46);(1,5,6,8)(2,10,7,3)(4,12,13,14)(9,11,15,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,38,39,41)(34,42,43,35)(36,46,37,48)(40,47,45,44)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gi,f^2=g,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, 10388153);