#291304

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fhi,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198500);