#7896030

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	#43504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4280 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)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,34)(28,37)(29,38)(30,33)(31,32)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(25,27,31,33)(26,30,32,34)(28,35,38,39)(29,36,37,40);(10,15)(11,17)(16,22)(18,23)(26,32)(27,33)(28,38)(36,40);(1,2,5,4)(3,6,8,7)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28)(26,29)(27,35)(30,36)(31,38)(32,37)(33,39)(34,40);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29)(26,28)(27,36)(30,35)(31,37)(32,38)(33,40)(34,39);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30)(26,33)(27,32)(28,36)(29,39)(31,34)(35,37)(38,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gi,e^2=gi,f^2=hi,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, 7803226);