#429572

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

989 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336768);