#428856

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
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

931 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,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=dei,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi

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, 336052);