#8603558

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

4205 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gh,e^2=gh,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, 8510754);