#7729105

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4408 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)(25,26,30,35)(27,31,37,39)(28,33,38,32)(29,34,36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24)(25,27,30,37)(26,31,35,39)(28,29,38,36)(32,40,33,34);(10,15)(12,16)(18,21)(20,23)(25,28,30,38)(26,32,35,33)(27,36,37,29)(31,34,39,40);(1,3,5,4)(2,7,6,8)(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20)(25,28)(26,33)(27,29)(30,38)(31,34)(32,35)(36,37)(39,40);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19)(25,26)(27,31)(28,33)(29,34)(30,35)(32,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)(25,29)(26,34)(27,28)(30,36)(31,33)(32,39)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ghi,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, 7636301);