#417546

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3027	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2464 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:

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

Polycyclic presentation:

2^9|a^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,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, 324742);