#215565

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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)(10,15)(11,18)(12,14)(16,23)(19,24)(20,22)(25,26,29,33,30,34,28,32)(27,35,36,31,38,39,37,40);(1,3,4,8)(2,5,6,7)(9,10,13,15)(11,19,20,16)(12,21,14,17)(18,23,22,24);(3,8)(5,7)(9,11,13,20)(10,16,15,19)(12,18,14,22)(17,23,21,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,35)(34,40);(2,6)(5,7)(9,12,13,14)(10,17,15,21)(11,18,20,22)(16,23,19,24)(25,28,30,29)(26,32,34,33)(27,36,38,37)(31,39,40,35)

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 122761);