#7726830

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4401 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)(9,10,14,16)(11,15,19,17)(12,13,18,23)(20,22,24,21)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,19)(16,17)(18,24)(21,23)(26,27)(29,37)(31,33)(35,40);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,18)(21,22)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(9,12,14,18)(10,13,16,23)(11,20,19,24)(15,22,17,21);(9,11)(10,17)(12,20)(13,21)(14,19)(15,16)(18,24)(22,23)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,13)(10,18)(11,21)(12,16)(14,23)(15,20)(17,24)(19,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,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, 7634026);