#224225

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

817 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)(6,7)(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,38)(39,40);(1,2,4,5)(3,7,8,6)(9,11,12,18)(10,14,15,17)(13,19,20,21)(16,22,23,24)(25,27)(26,30)(28,35)(29,36)(31,39)(32,34)(33,38)(37,40);(10,15)(11,18)(13,20)(22,24)(26,31)(27,35)(29,37)(30,38)(32,33)(34,39);(1,3,4,8)(2,6,5,7)(11,18)(14,17)(19,21)(22,24)(27,35)(30,34)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131421);