#8772607

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3997 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

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, 8679803);