#426699

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333895);