#8739427

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

3933 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)(3,10)(4,6)(5,8)(7,9)(11,12)(13,15)(14,16)(17,18,22,27)(19,25,28,23)(20,24,29,30)(21,26,31,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,48)(44,45)(46,47);(1,3,7,5)(2,8,9,10)(4,13,14,11)(6,12,16,15)(17,19)(18,23)(20,26)(21,24)(22,28)(25,27)(29,32)(30,31)(33,35,39,38)(34,40,42,43)(36,46,41,44)(37,45,48,47);(2,9)(4,14)(8,10)(11,13)(17,20,22,29)(18,24,27,30)(19,26,28,32)(21,25,31,23)(33,36,39,41)(34,37,42,48)(35,44,38,46)(40,47,43,45);(1,4,7,14)(2,6,9,16)(3,11,5,13)(8,15,10,12)(17,20)(18,24)(19,26)(21,23)(22,29)(25,31)(27,30)(28,32)(33,37)(34,41)(35,45)(36,42)(38,47)(39,48)(40,44)(43,46);(1,5,7,3)(2,10,9,8)(4,13,14,11)(6,12,16,15)(17,19)(18,25)(20,26)(21,30)(22,28)(23,27)(24,31)(29,32);(1,6)(2,4)(3,12)(5,15)(7,16)(8,13)(9,14)(10,11)(17,21,22,31)(18,26,27,32)(19,24,28,30)(20,25,29,23)(33,38,39,35)(34,40,42,43)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,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, 8646623);