#8772473

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

4128 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,15)(14,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,34,38,42)(35,40,45,39)(36,46,41,37)(43,48,47,44);(2,7)(5,14)(8,11)(9,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34)(35,39)(36,37)(38,42)(40,45)(41,46)(43,48)(44,47);(3,12)(4,13)(8,11)(9,15)(18,22)(19,26)(24,28)(27,32)(33,35,38,45)(34,39,42,40)(36,47,41,43)(37,44,46,48);(1,3,6,12)(2,8,7,11)(4,10,13,16)(5,9,14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,38,41)(34,37,42,46)(35,43,45,47)(39,48,40,44);(1,4,6,13)(2,9,7,15)(3,10,12,16)(5,8,14,11)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,42)(41,47)(46,48);(1,5,6,14)(2,10,7,16)(3,9,12,15)(4,8,13,11)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,41)(35,44)(36,42)(38,46)(39,43)(40,47)(45,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=g,e^2=gh,f^2=gh,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, 8679669);