#8810084

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

3914 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717280);