#8728408

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

4257 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)(3,4)(5,9)(6,7)(8,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,30)(28,29)(31,32)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(2,7)(3,8)(9,16)(10,13)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(2,7)(3,8)(9,16)(10,13)(19,26)(22,23)(27,32)(30,31)(36,44)(41,46)(43,45)(47,48);(1,3,6,8)(2,4,7,12)(5,13,15,10)(9,14,16,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,30)(29,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4)(2,8)(3,7)(5,14)(6,12)(9,10)(11,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32)(33,37,39,35)(34,40,38,42)(36,45,44,43)(41,47,46,48);(1,5,6,15)(2,9,7,16)(3,10,8,13)(4,11,12,14)(17,18)(19,22)(20,28)(21,25)(23,26)(24,29)(27,31)(30,32)(33,38,39,34)(35,42,37,40)(36,41,44,46)(43,47,45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635604);