#8258634

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8165830);