#8107211

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

4119 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(17,18)(19,20)(21,29)(22,23)(24,26)(25,30)(27,32)(28,31)(34,39)(35,43)(37,41)(44,48);(2,7)(3,10)(8,13)(11,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,46)(44,47)(45,48);(2,7)(3,10)(8,13)(11,16)(18,23)(19,24)(25,29)(27,31)(35,43)(40,46)(44,48)(45,47);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,18)(19,26)(20,24)(21,25)(22,23)(27,32)(28,31)(29,30)(33,35,38,43)(34,40,39,46)(36,44,42,48)(37,45,41,47);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,20,22,26)(18,24,23,19)(21,28,30,32)(25,31,29,27)(33,36,38,42)(34,41,39,37)(35,44,43,48)(40,47,46,45);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)(33,37,38,41)(34,42,39,36)(35,45,43,47)(40,48,46,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014407);