#9851098

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	#56381	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3980 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=gh,e^2=i,f^2=g,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, 9758294);