#8925310

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

4306 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^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, 8832506);