#8175726

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gh,d^2=g,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8082922);