#8826288

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	#55094	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,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=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8733484);