#8097738

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

4155 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004934);