#395583

order	512
nilpotency class	3
isoclinism	isoclinic class of #395581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2185 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,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, 302779);