#7981193

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7888389);