#8095786

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

3845 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)(19,26,29,24)(20,25,22,30)(27,32,28,31)(33,34)(35,37)(36,40)(38,41)(39,43)(42,45)(44,47)(46,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,19)(18,24)(20,27)(21,26)(22,28)(23,29)(25,31)(30,32)(33,35,38,43)(34,37,41,39)(36,44,46,42)(40,47,48,45);(2,7)(4,8)(10,14)(12,15)(17,20,23,22)(18,25,21,30)(19,27,29,28)(24,31,26,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,46)(45,48);(1,3,6,9)(2,8,7,4)(5,11,13,16)(10,15,14,12)(17,19)(18,24)(20,27)(21,26)(22,28)(23,29)(25,31)(30,32)(33,36,38,46)(34,40,41,48)(35,42,43,44)(37,45,39,47);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,18)(19,26,29,24)(20,25,22,30)(27,31,28,32);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,22,23,20)(18,25,21,30)(19,28,29,27)(24,31,26,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,48,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=i,e^2=ghi,f^2=gh,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, 8002982);