#8757094

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,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, 8664290);