#8458488

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=i,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, 8365684);