#7977605

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

4481 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=g,f^2=h,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, 7884801);