#10325577

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

4305 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)(3,10)(4,9)(5,14)(6,7)(8,11)(12,13)(15,16)(18,22)(23,26)(24,28)(30,32)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,7)(3,11)(5,9)(12,16)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30)(33,34,38,36)(35,41,43,39)(37,44,45,40)(42,46,48,47);(2,7)(4,14)(8,10)(12,16)(19,25)(23,26)(27,31)(30,32)(34,36)(39,41)(40,44)(46,47);(1,3)(2,8)(4,12)(5,13)(6,11)(7,10)(9,15)(14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,15)(10,13)(11,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5,6,9)(2,4,7,14)(3,13,11,15)(8,12,10,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10232773);