#8932099

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	#36818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,12)(15,16)(17,18,23,27)(19,24,26,21)(20,29,22,25)(28,32,30,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,46)(42,47)(44,48);(1,3,6,7)(2,4,9,10)(5,11,14,15)(8,12,13,16)(17,19,23,26)(18,24,27,21)(20,30,22,28)(25,32,29,31)(34,39)(36,40)(41,45)(44,47);(1,2,6,9)(3,4,7,10)(5,13,14,8)(11,16,15,12)(17,20,23,22)(18,25,27,29)(19,28,26,30)(21,32,24,31)(33,34,38,39)(35,40,43,36)(37,41,46,45)(42,47,48,44);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,21,23,24)(18,26,27,19)(20,31,22,32)(25,28,29,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,45)(46,48);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,22,23,20)(18,25,27,29)(19,28,26,30)(21,31,24,32)(33,36,38,40)(34,35,39,43)(37,44,46,47)(41,42,45,48);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,18)(19,24)(20,29)(21,26)(22,25)(23,27)(28,32)(30,31)(33,37,38,46)(34,41,39,45)(35,42,43,48)(36,44,40,47)

Polycyclic presentation:

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

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, 8839295);