#8755114

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	#46052	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3841 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gi,e^2=h,f^2=g,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, 8662310);