#8821818

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	#53993	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4051 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=gh,e^2=ghi,f^2=hi,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, 8729014);