#8739331

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

4075 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)(3,5)(4,11)(6,7)(8,13)(9,15)(10,12)(14,16)(17,18,22,27)(19,25,28,23)(20,24,29,30)(21,26,31,32)(35,37)(36,40)(43,45)(44,46);(2,7)(3,9)(8,11)(10,16)(17,19)(18,23)(20,26)(21,24)(22,28)(25,27)(29,32)(30,31)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(2,7)(3,9)(8,11)(10,16)(17,20,22,29)(18,24,27,30)(19,26,28,32)(21,25,31,23)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,3,6,9)(2,5,7,15)(4,10,13,16)(8,14,11,12)(17,20)(18,24)(19,26)(21,23)(22,29)(25,31)(27,30)(28,32)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,4,6,13)(2,8,7,11)(3,10,9,16)(5,14,15,12)(17,19)(18,25)(20,26)(21,30)(22,28)(23,27)(24,31)(29,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,11)(13,14)(17,21,22,31)(18,26,27,32)(19,24,28,30)(20,25,29,23)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 8646527);