#8184832

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	#40008	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4079 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)(3,5)(4,7)(6,14)(8,10)(9,16)(11,12)(13,15)(17,18,23,22)(19,26,21,24)(20,25,28,30)(27,32,29,31)(33,34,38,42)(35,40,36,39)(37,41,45,48)(43,47,44,46);(1,3,8,7)(2,5,10,4)(6,11,16,15)(9,13,14,12)(20,28)(25,30)(27,29)(31,32)(33,35,38,36)(34,39,42,40)(37,43,45,44)(41,46,48,47);(1,4,8,5)(2,3,10,7)(6,12,16,13)(9,11,14,15)(17,19,23,21)(18,24,22,26)(20,27,28,29)(25,31,30,32)(33,35,38,36)(34,39,42,40)(37,44,45,43)(41,47,48,46);(1,5,8,4)(2,3,10,7)(6,13,16,12)(9,11,14,15)(17,20,23,28)(18,25,22,30)(19,27,21,29)(24,31,26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6,8,16)(2,9,10,14)(3,11,7,15)(4,12,5,13)(17,21,23,19)(18,26,22,24)(20,29,28,27)(25,32,30,31)(33,36,38,35)(34,40,42,39)(37,44,45,43)(41,47,48,46);(1,7,8,3)(2,4,10,5)(6,15,16,11)(9,12,14,13)(17,22,23,18)(19,26,21,24)(20,30,28,25)(27,32,29,31)(33,37,38,45)(34,41,42,48)(35,43,36,44)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092028);