#10407819

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

4267 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(33,34,39,43)(35,42,46,40)(36,38,37,41)(44,45,47,48);(2,3)(5,13)(7,9)(12,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,35)(34,40)(36,47)(37,44)(38,45)(39,46)(41,48)(42,43);(4,11)(7,9)(12,16)(14,15)(18,20)(23,25)(24,28)(30,31)(33,36,39,37)(34,41,43,38)(35,44,46,47)(40,45,42,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,37,39,36)(34,38,43,41)(35,44,46,47)(40,45,42,48);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,35)(34,42)(36,44)(37,47)(38,45)(39,46)(40,43)(41,48);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,38)(34,36)(35,45)(37,43)(39,41)(40,47)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,d^2=ghi,e^2=hi,f^2=hi,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, 10315015);