#8533407

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

4199 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8440603);