#7893705

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	#38749	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4377 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7800901);