#9863035

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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)(3,10)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18)(19,27)(20,30)(21,24)(22,26)(23,25)(28,31)(29,32)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,48,46);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(17,19,24,23)(18,25,21,27)(20,29,22,28)(26,32,30,31)(35,40)(37,42)(43,48)(45,46);(17,20,24,22)(18,26,21,30)(19,28,23,29)(25,31,27,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,21,24,18)(19,27,23,25)(20,26,22,30)(28,31,29,32)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,5,6,10)(2,8,9,3)(4,13,14,16)(7,15,12,11)(17,22,24,20)(18,26,21,30)(19,29,23,28)(25,31,27,32)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,23,24,19)(18,27,21,25)(20,29,22,28)(26,32,30,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9770231);