#8720708

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	#53959	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627904);