#8755165

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	#54588	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=gi,e^2=gh,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, 8662361);