#7736698

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

4488 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)(9,10,14,19)(11,16,20,15)(12,17,22,23)(13,18,24,21)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,24)(18,22)(25,26,31,29)(27,34,28,32)(30,37,38,33)(35,39,36,40);(11,20)(12,22)(15,16)(17,23)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,4,5,3)(2,8,6,7)(9,12,14,22)(10,17,19,23)(11,21,20,18)(13,15,24,16)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,13)(10,18)(11,17)(12,15)(14,24)(16,22)(19,21)(20,23)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643894);