#8755200

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

3981 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,18,22,25)(19,20,27,23)(21,29,30,24)(26,31,32,28)(34,38)(39,44)(40,45)(46,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28)(36,43)(40,45)(41,47)(46,48);(1,2,5,8)(3,6,11,9)(4,7,13,12)(10,14,16,15)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,30)(31,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31)(33,35,37,42)(34,39,38,44)(36,41,43,47)(40,46,45,48);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36,37,43)(34,40,38,45)(35,41,42,47)(39,46,44,48);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,21,22,30)(18,24,25,29)(19,26,27,32)(20,28,23,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,44)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662396);