#8920727

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

4208 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8827923);