#8436230

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

4028 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=g,f^2=g,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, 8343426);