#8434153

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

4212 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)(3,13)(4,10)(5,7)(6,15)(8,12)(9,16)(11,14)(20,28)(24,31)(27,32)(29,30)(33,34,40,38)(35,44,46,41)(36,47,42,37)(39,43,48,45);(1,3)(2,9)(4,6)(5,14)(7,15)(8,16)(10,11)(12,13)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35,40,46)(34,41,38,44)(36,39,42,48)(37,43,47,45);(1,4,8,5)(2,10,12,7)(3,6,16,14)(9,11,13,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,37)(35,39)(38,47)(40,42)(41,43)(44,45)(46,48);(1,5,8,4)(2,10,12,7)(3,14,16,6)(9,11,13,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,40,47)(34,42,38,36)(35,45,46,43)(39,44,48,41);(1,6,8,14)(2,11,12,15)(3,4,16,5)(7,9,10,13)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,38,40,34)(35,44,46,41)(36,47,42,37)(39,45,48,43);(1,7,8,10)(2,5,12,4)(3,15,16,11)(6,9,14,13)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,39,40,48)(34,43,38,45)(35,42,46,36)(37,41,47,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,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, 8341349);