#8096568

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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)(3,11)(4,7)(5,9)(6,10)(8,13)(12,14)(15,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34,39,42)(35,37,40,36)(38,41,46,48)(43,45,47,44);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,15,14,10)(20,28)(24,31)(27,32)(29,30)(33,35,39,40)(34,36,42,37)(38,43,46,47)(41,44,48,45);(1,2,7,4)(3,11,13,8)(5,9,12,14)(6,10,16,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,37)(34,35,42,40)(38,44,46,45)(41,43,48,47);(1,4,7,2)(3,8,13,11)(5,14,12,9)(6,15,16,10)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5)(2,9)(3,6)(4,14)(7,12)(8,15)(10,11)(13,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,39,36)(34,40,42,35)(38,44,46,45)(41,43,48,47);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,9)(11,14)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,46)(34,41,42,48)(35,43,40,47)(36,44,37,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003764);