#8493295

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

4043 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,13)(14,15)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34,38,42)(35,41,45,39)(36,37,40,46)(43,48,47,44);(1,3,7,4)(2,8,11,9)(5,14,16,12)(6,13,10,15)(19,28)(21,29)(24,32)(25,30)(33,34,38,42)(35,41,45,39)(36,46,40,37)(43,44,47,48);(1,3,7,4)(2,9,11,8)(5,12,16,14)(6,13,10,15)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(1,4,7,3)(2,9,11,8)(5,14,16,12)(6,13,10,15)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36,38,40)(34,37,42,46)(35,43,45,47)(39,44,41,48);(1,5)(2,10)(3,12)(4,14)(6,11)(7,16)(8,13)(9,15)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37)(34,40)(35,44)(36,42)(38,46)(39,47)(41,43)(45,48);(1,6,7,10)(2,5,11,16)(3,13,4,15)(8,14,9,12)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,35)(34,41)(36,47)(37,44)(38,45)(39,42)(40,43)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8400491);