#8458127

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

4179 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(18,23)(19,28)(26,30)(27,32)(33,34,39,43)(35,40,46,44)(36,38,42,48)(37,47,45,41);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(33,34,39,43)(35,44,46,40)(36,38,42,48)(37,41,45,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19)(18,24)(20,27)(21,26)(22,28)(23,25)(29,32)(30,31)(33,35,39,46)(34,40,43,44)(36,45,42,37)(38,41,48,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,19)(18,25)(20,27)(21,30)(22,28)(23,24)(26,31)(29,32)(33,36)(34,38)(35,45)(37,46)(39,42)(40,41)(43,48)(44,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,20,22,29)(18,26,23,30)(19,27,28,32)(21,25,31,24)(33,37,39,45)(34,41,43,47)(35,36,46,42)(38,40,48,44);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,21,22,31)(18,27,23,32)(19,26,28,30)(20,25,29,24)(33,38)(34,42)(35,41)(36,43)(37,40)(39,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8365323);