#7986332

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4351 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)(9,10,15,17)(11,16,12,14)(13,21,22,18)(19,24,20,23)(25,26,30,29)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,17,14)(13,19,22,20)(18,23,21,24)(26,29)(31,33)(32,36)(38,39);(10,17)(11,12)(13,22)(23,24)(25,26,30,29)(27,31,35,33)(28,36,37,32)(34,39,40,38);(9,12,15,11)(10,16,17,14)(13,20,22,19)(18,23,21,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(1,4,5,2)(3,7,8,6)(9,13,15,22)(10,18,17,21)(11,19,12,20)(14,24,16,23)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(9,14,15,16)(10,11,17,12)(13,23,22,24)(18,20,21,19)(25,29,30,26)(27,33,35,31)(28,32,37,36)(34,38,40,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=h,e^2=gh,f^2=i,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, 7893528);