#8755400

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	#46093	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3833 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 8662596);