#8621398

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	#49984	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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)(3,8)(4,10)(5,14)(6,12)(7,9)(11,13)(15,16)(17,18,23,26)(19,21,24,29)(20,22,25,31)(27,28,30,32)(33,34,40,39)(35,44,38,41)(36,43,37,42)(45,47,46,48);(1,3)(2,8)(4,13)(5,15)(6,9)(7,12)(10,11)(14,16)(17,19,23,24)(18,21,26,29)(20,27,25,30)(22,28,31,32)(33,35,40,38)(34,41,39,44)(36,46,37,45)(42,48,43,47);(2,9)(3,12)(5,13)(10,16)(19,24)(21,29)(27,30)(28,32)(33,36,40,37)(34,42,39,43)(35,45,38,46)(41,47,44,48);(1,4)(2,10)(3,5)(6,11)(7,15)(8,14)(9,16)(12,13)(17,20)(18,22)(19,27)(21,28)(23,25)(24,30)(26,31)(29,32)(33,37,40,36)(34,43,39,42)(35,46,38,45)(41,48,44,47);(1,5)(2,11)(3,4)(6,10)(7,13)(8,16)(9,14)(12,15)(17,21,23,29)(18,24,26,19)(20,28,25,32)(22,30,31,27)(33,38,40,35)(34,44,39,41)(36,46,37,45)(42,48,43,47);(1,6)(2,12)(3,9)(4,14)(5,10)(7,8)(11,15)(13,16)(17,22)(18,25)(19,28)(20,26)(21,30)(23,31)(24,32)(27,29)(33,39,40,34)(35,44,38,41)(36,43,37,42)(45,48,46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8528594);