#8720375

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	#59977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=hi,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, 8627571);