#8508879

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(3,12)(4,10)(5,9)(6,8)(7,11)(13,15)(14,16)(17,18,23,22)(19,27,29,24)(20,25,31,30)(21,26,28,32)(33,34)(35,41)(36,44)(37,38)(39,40)(42,46)(43,45)(47,48);(1,3,7,6)(2,8,11,12)(4,13,5,14)(9,15,10,16)(17,18,23,22)(19,27,29,24)(20,30,31,25)(21,32,28,26)(34,40)(37,41)(42,44)(45,48);(1,4,7,5)(2,9,11,10)(3,13,6,14)(8,15,12,16)(17,19,23,29)(18,24,22,27)(20,28,31,21)(25,26,30,32)(33,35,39,38)(34,41,40,37)(36,43,46,47)(42,48,44,45);(1,5,7,4)(2,10,11,9)(3,13,6,14)(8,15,12,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,31)(24,26)(27,32)(33,36)(34,42)(35,43)(37,45)(38,47)(39,46)(40,44)(41,48);(1,2)(3,12)(4,9)(5,10)(6,8)(7,11)(13,16)(14,15)(17,21,23,28)(18,26,22,32)(19,20,29,31)(24,30,27,25)(33,37)(34,35)(36,45)(38,40)(39,41)(42,43)(44,47)(46,48);(1,6,7,3)(2,8,11,12)(4,14,5,13)(9,15,10,16)(17,22,23,18)(19,27,29,24)(20,25,31,30)(21,32,28,26)(33,38,39,35)(34,37,40,41)(36,43,46,47)(42,48,44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8416075);