#395695

order	512
nilpotency class	3
isoclinism	isoclinic class of #395693
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1882 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)(5,6,11,15,10,16,9,14)(7,17,25,28,20,27,19,12)(8,13,22,29,24,30,23,31)(18,32,33,26,21,35,34,36);(5,7,9,19,10,20,11,25)(6,12,15,17,16,28,14,27)(8,21,23,33,24,18,22,34)(13,26,29,35,30,36,31,32);(5,8)(6,13)(7,18)(9,22)(10,24)(11,23)(12,26)(14,29)(15,31)(16,30)(17,32)(19,33)(20,21)(25,34)(27,35)(28,36);(2,4)(5,9,10,11)(6,14,16,15)(7,19,20,25)(8,22,24,23)(12,27,28,17)(13,29,30,31)(18,33,21,34)(26,35,36,32)

Polycyclic presentation:

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

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, 302891);