#215447

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	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1004 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)(11,17,18,15)(12,20)(13,14)(16,21)(19,22,24,23)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,38)(39,40);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(12,21)(22,23)(25,27)(26,30)(28,35)(29,36)(31,39)(32,34)(33,38)(37,40);(3,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(26,31)(27,35)(29,37)(30,38)(32,33)(34,39);(2,6)(5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(26,32)(30,34)(31,33)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,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, 122643);