#428061

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

1145 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:

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

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335257);