#10534394

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

4286 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=g,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10441590);