#10202092

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
prime radical	#53693	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

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

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, 10109288);