#412956

order	512
nilpotency class	3
isoclinism	isoclinic class of #396715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi

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, 320152);