#7726824

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	#47934	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4593 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=ghi,f^2=g,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, 7634020);