#7726803

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	#37439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4728 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^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, 7633999);