#8549581

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	#54364	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

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, 8456777);