#424612

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1255 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)(7,15,18,11)(8,12)(9,14)(10,13)(16,22,26,19)(17,25,33,27)(20,29)(21,28)(23,31)(24,30)(32,35,36,34);(5,7)(6,11)(8,17)(9,22)(10,18)(12,27)(13,15)(14,16)(19,24)(20,35)(21,33)(23,34)(25,28)(26,30)(29,32)(31,36);(5,6)(7,16,18,26)(8,12)(9,14)(10,13)(11,22,15,19)(17,32,33,36)(20,29)(21,28)(23,31)(24,30)(25,34,27,35);(2,4)(5,8,10,21)(6,12,13,28)(7,17,18,33)(9,23,24,20)(11,25,15,27)(14,31,30,29)(16,32,26,36)(19,35,22,34)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,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, 331808);