#8757353

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

3977 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,18,22,23)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34)(35,41)(36,44)(37,42)(38,39)(40,46)(43,45)(47,48);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(18,23)(24,26)(25,28)(31,32)(34,39)(35,42)(40,44)(43,48);(1,2,5,8)(3,6,11,9)(4,7,13,12)(10,14,16,15)(17,19,22,21)(18,24,23,26)(20,29,30,27)(25,32,28,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,20,22,30)(18,25,23,28)(19,27,21,29)(24,31,26,32)(33,36,38,46)(34,40,39,44)(35,43,42,48)(37,45,41,47);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,21,22,19)(18,26,23,24)(20,29,30,27)(25,32,28,31)(33,34)(35,37)(36,40)(38,39)(41,42)(43,45)(44,46)(47,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,37)(34,35)(36,45)(38,41)(39,42)(40,43)(44,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664549);