#224937

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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

785 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132133);