#375690

order	512
nilpotency class	3
isoclinism	isoclinic class of #213778
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=fh,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282886);