#7726807

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	#36739	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4630 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)(9,10,14,12)(11,17,18,15)(13,16,22,20)(19,24,21,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,17)(13,21)(14,18)(16,24)(19,22)(20,23)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,39,40);(10,12)(11,18)(16,20)(19,21)(27,29)(31,34)(35,36)(38,40);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,17,18,15)(13,20,22,16)(19,24,21,23)(25,28,30,37)(26,32,33,39)(27,35,29,36)(31,38,34,40);(9,11)(10,15)(12,17)(13,19)(14,18)(16,23)(20,24)(21,22)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,39,40);(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,24)(18,21)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7634003);