#8855451

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

Polycyclic presentation:

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

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, 8762647);