#8445183

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

4169 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(20,28)(24,31)(27,32)(29,30)(33,34,39,44)(35,45,47,40)(36,37,42,41)(38,43,46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,42)(34,41,44,37)(35,46,47,38)(40,48,45,43);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,34)(35,45)(36,41)(37,42)(38,48)(39,44)(40,47)(43,46);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,41)(34,42,44,36)(35,43,47,48)(38,45,46,40);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,46)(34,43,44,48)(35,42,47,36)(37,45,41,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,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, 8352379);