#214573

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121769);