#401063

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1717 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:

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

Polycyclic presentation:

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

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, 308259);