#217224

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

738 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124420);