#10320436

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4755 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227632);