#8893726

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

4216 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8800922);