#8650386

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	#53115	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=ghi,e^2=g,f^2=gi,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, 8557582);