#413093

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1528 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,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, 320289);