#8755365

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	#53673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,18,23,21)(19,20,26,24)(22,29,30,25)(27,31,32,28)(33,34,39,36)(35,38,43,40)(37,41,45,44)(42,46,48,47);(2,7)(4,8)(5,13)(10,16)(18,21)(19,26)(25,29)(27,32)(37,45)(41,44)(42,48)(46,47);(1,3,6,11)(2,8,7,4)(5,10,13,16)(9,15,14,12)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35,39,43)(34,40,36,38)(37,42,45,48)(41,47,44,46);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,36,39,34)(35,38,43,40)(37,44,45,41)(42,46,48,47);(1,4)(2,3)(5,12)(6,8)(7,11)(9,10)(13,15)(14,16)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31)(33,37,39,45)(34,41,36,44)(35,42,43,48)(38,46,40,47);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38)(34,35)(36,43)(37,46)(39,40)(41,42)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=hi,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, 8662561);