#8762016

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

4294 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)(3,5)(4,12)(6,8)(7,11)(9,13)(10,15)(14,16)(17,18)(19,22)(20,28)(21,24)(23,29)(25,27)(26,31)(30,32)(33,34,38,42)(35,36,40,39)(37,41,46,48)(43,44,45,47);(1,3,6,11)(2,7,8,5)(4,10,13,16)(9,15,12,14)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,30)(31,32)(33,35,38,40)(34,39,42,36)(37,45,46,43)(41,44,48,47);(2,8)(4,13)(5,7)(10,16)(17,18,21,24)(19,25,27,22)(20,23,29,28)(26,31,32,30)(33,34,38,42)(35,39,40,36)(37,41,46,48)(43,47,45,44);(1,4)(2,9)(3,10)(5,14)(6,13)(7,15)(8,12)(11,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,29)(31,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,5,6,7)(2,3,8,11)(4,14,13,15)(9,10,12,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32)(33,36,38,39)(34,40,42,35)(37,44,46,47)(41,45,48,43);(1,3)(2,7)(4,10)(5,8)(6,11)(9,15)(12,14)(13,16)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,31)(27,32)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,45)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669212);