#8463692

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370888);