#472560

order	512
nilpotency class	3
isoclinism	isoclinic class of #457592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379756);