#8925293

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

4048 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8832489);