#8445396

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=h,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8352592);