#8549476

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,e^2=h,f^2=hi,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, 8456672);