#8761989

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

4284 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,15)(14,16)(17,18,22,25)(19,23,28,24)(20,21,30,32)(26,27,29,31)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(2,7)(5,14)(8,11)(9,15)(17,19)(18,23)(20,29)(21,31)(22,28)(24,25)(26,30)(27,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,48)(45,46);(3,12)(4,13)(8,11)(9,15)(17,19,22,28)(18,24,25,23)(20,29,30,26)(21,27,32,31)(33,34,39,38)(35,40,37,42)(36,44,46,41)(43,48,45,47);(1,3,6,12)(2,8,7,11)(4,10,13,16)(5,9,14,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,4,6,13)(2,9,7,15)(3,10,12,16)(5,8,14,11)(17,20)(18,21)(19,26)(22,30)(23,27)(24,31)(25,32)(28,29)(33,37,39,35)(34,40,38,42)(36,45,46,43)(41,47,44,48);(1,5,6,14)(2,10,7,16)(3,9,12,15)(4,8,13,11)(17,21)(18,20)(19,27)(22,32)(23,26)(24,29)(25,30)(28,31)(33,38,39,34)(35,40,37,42)(36,44,46,41)(43,47,45,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=g,f^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, 8669185);