#7896028

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	#44885	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26,31,33)(27,32,36,39)(28,38,34,29)(30,35,37,40);(1,3,5,8)(2,6,4,7)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27)(26,32)(28,37)(29,35)(30,34)(31,36)(33,39)(38,40);(10,17)(12,20)(16,19)(21,23)(26,33)(27,36)(28,34)(35,40);(1,2,5,4)(3,6,8,7)(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,28,31,34)(26,29,33,38)(27,37,36,30)(32,35,39,40);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,29)(26,34)(27,35)(28,33)(30,32)(31,38)(36,40)(37,39);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,30,31,37)(26,35,33,40)(27,34,36,28)(29,39,38,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=hi,f^2=gi,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, 7803224);