#8458078

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

4195 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=i,d^2=gh,e^2=hi,f^2=h,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, 8365274);