#8551254

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	#55414	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8458450);