#10325623

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

4318 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^2=h,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, 10232819);