#8125485

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

4064 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=hi,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8032681);