#8095783

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002979);