#10349021

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

4206 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)(3,12)(4,8)(5,9)(6,13)(7,11)(10,16)(14,15)(18,23)(19,25)(21,29)(28,31)(33,34,38,41)(35,36,39,44)(37,40,46,48)(42,43,45,47);(3,4)(6,16)(8,12)(10,13)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,35,38,39)(34,36,41,44)(37,45,46,42)(40,47,48,43);(1,3,7,4)(2,8,11,12)(5,10,15,13)(6,14,16,9)(17,19)(18,20)(21,27)(22,25)(23,24)(26,28)(29,32)(30,31)(33,36,38,44)(34,39,41,35)(37,43,46,47)(40,45,48,42);(1,4,7,3)(2,8,11,12)(5,10,15,13)(6,9,16,14)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,30)(29,32)(33,35)(34,36)(37,42)(38,39)(40,43)(41,44)(45,46)(47,48);(1,5,7,15)(2,9,11,14)(3,13,4,10)(6,8,16,12)(17,20,22,24)(18,25,23,19)(21,28,29,31)(26,32,30,27)(33,36)(34,39)(35,41)(37,43)(38,44)(40,45)(42,48)(46,47);(1,6,7,16)(2,10,11,13)(3,14,4,9)(5,8,15,12)(17,21,22,29)(18,26,23,30)(19,27,25,32)(20,28,24,31)(33,37)(34,40)(35,42)(36,43)(38,46)(39,45)(41,48)(44,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=gh,e^2=h,f^2=h,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, 10256217);