#8425445

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

4256 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

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, 8332641);