#8739432

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	#53610	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,23,27)(19,28,30,24)(20,26,31,25)(21,32,29,22)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19)(18,24)(20,21)(22,26)(23,30)(25,32)(27,28)(29,31)(34,39)(35,42)(41,44)(43,48);(4,12)(8,15)(11,13)(14,16)(17,20,23,31)(18,25,27,26)(19,21,30,29)(22,24,32,28)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,4,6,12)(2,8,9,15)(3,11,5,13)(7,14,10,16)(17,20)(18,26)(19,29)(21,30)(22,28)(23,31)(24,32)(25,27)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21,23,29)(18,22,27,32)(19,20,30,31)(24,26,28,25);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,22)(18,21)(19,26)(20,24)(23,32)(25,30)(27,29)(28,31)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8646628);