#448932

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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)(5,6,9,16,11,15,10,14)(7,17,19,28,21,27,20,12)(8,13,25,30,24,32,23,31)(18,33,22,36,35,29,34,26);(5,7,9,19,11,21,10,20)(6,12,16,17,15,28,14,27)(8,22,25,35,24,34,23,18)(13,29,30,26,32,33,31,36);(5,8,10,25,11,24,9,23)(6,13,16,31,15,32,14,30)(7,18,19,34,21,35,20,22)(12,26,27,33,28,36,17,29);(2,4)(7,19,21,20)(12,27,28,17)(18,34,35,22)(26,33,36,29)

Polycyclic presentation:

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

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, 356128);