#8920725

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

4135 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)(3,10)(4,9)(5,14)(6,7)(8,11)(12,16)(13,15)(17,18,23,25)(19,24,29,31)(20,26,30,32)(21,27,28,22)(33,34,39,37)(35,40,42,38)(36,41,44,46)(43,47,48,45);(2,7)(4,14)(8,10)(12,15)(17,19)(18,24)(20,28)(21,30)(22,26)(23,29)(25,31)(27,32)(33,35,39,42)(34,40,37,38)(36,43,44,48)(41,47,46,45);(2,7)(3,11)(5,9)(12,15)(18,25)(19,29)(20,30)(22,27)(35,42)(36,44)(38,40)(41,46);(1,3,6,11)(2,8,7,10)(4,15,14,12)(5,13,9,16)(17,20,23,30)(18,26,25,32)(19,28,29,21)(22,31,27,24)(33,36)(34,41)(35,43)(37,46)(38,45)(39,44)(40,47)(42,48);(1,4)(2,9)(3,12)(5,7)(6,14)(8,13)(10,16)(11,15)(17,21,23,28)(18,27,25,22)(19,20,29,30)(24,26,31,32)(33,37,39,34)(35,38,42,40)(36,41,44,46)(43,47,48,45);(1,5,6,9)(2,4,7,14)(3,13,11,16)(8,15,10,12)(17,22,23,27)(18,28,25,21)(19,26,29,32)(20,31,30,24)(33,38)(34,35)(36,45)(37,42)(39,40)(41,43)(44,47)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,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, 8827921);