#8471026

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 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)(3,13)(4,10)(5,7)(6,16)(8,12)(9,15)(11,14)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28)(33,34,39,36)(35,40,46,43)(37,47,45,41)(38,42,44,48);(1,3,8,15)(2,9,12,13)(4,14,5,6)(7,11,10,16)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,42)(47,48);(1,4,8,5)(2,10,12,7)(3,14,15,6)(9,16,13,11)(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,34,39,36)(35,43,46,40)(37,41,45,47)(38,42,44,48);(1,5,8,4)(2,10,12,7)(3,6,15,14)(9,16,13,11)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30)(33,36,39,34)(35,43,46,40)(37,47,45,41)(38,42,44,48);(1,6)(2,11)(3,4)(5,15)(7,13)(8,14)(9,10)(12,16)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31)(33,37,39,45)(34,41,36,47)(35,44,46,38)(40,42,43,48);(1,7,8,10)(2,5,12,4)(3,11,15,16)(6,13,14,9)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)(33,38,39,44)(34,42,36,48)(35,45,46,37)(40,41,43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=ghi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8378222);