#7749858

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	#42952	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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)(3,5,7,8)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26,29,30)(27,33,35,31)(28,32,36,37)(34,39,40,38);(1,2)(3,5)(4,6)(7,8)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(10,11)(15,17)(16,18)(22,23)(26,30)(28,36)(31,33)(34,40);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3,4,7)(2,5,6,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7657054);