#292249

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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 ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

846 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)(9,10,15,11)(12,20,21,24)(13,19,14,17)(16,23,18,22)(25,26)(27,34)(28,32)(29,39)(30,33)(31,37)(35,40)(36,38);(1,3,4,8)(2,5,6,7)(9,11,15,10)(12,20,21,24)(13,19,14,17)(16,22,18,23)(25,27)(26,31)(28,38)(29,36)(30,37)(32,40)(33,34)(35,39);(9,12,14,23,15,21,13,22)(10,16,19,24,11,18,17,20)(25,28,30,39)(26,32,33,29)(27,35)(31,40)(34,36)(37,38);(2,6)(5,7)(12,21)(16,18)(20,24)(22,23)(25,29,30,32)(26,28,33,39)(27,36,37,40)(31,38,34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fhi,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 199445);