#399125

order	512
nilpotency class	3
isoclinism	isoclinic class of #399124
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1290 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)(7,17,20,32)(8,13,24,31)(9,16,11,14)(12,25,27,19)(18,33,34,28)(21,36,35,26)(22,30,23,29);(5,7,11,25,10,20,9,19)(6,12,16,32,15,27,14,17)(8,21,22,34,24,35,23,18)(13,28,29,36,31,33,30,26);(5,8,11,22,10,24,9,23)(6,13,16,29,15,31,14,30)(7,18,25,21,20,34,19,35)(12,26,32,28,27,36,17,33);(2,4)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ef,e^2=i,g^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,g^c=gi

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, 306321);