#8012025

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

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7919221);