#8757053

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

4133 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664249);