#470219

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

745 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)(5,6,11,16,12,17,10,28)(7,18,9,15,21,30,20,13)(8,22,24,14,26,36,25,33)(19,29,34,35,23,31,27,32);(5,7,10,20,12,21,11,9)(6,13,16,18,17,15,28,30)(8,19,24,34,26,23,25,27)(14,31,22,35,33,29,36,32);(5,8)(6,14,17,33)(7,19,21,23)(9,27,20,34)(10,24)(11,25)(12,26)(13,29)(15,31)(16,22,28,36)(18,32)(30,35);(2,4)(5,9,11,21,12,20,10,7)(6,15,16,30,17,13,28,18)(8,23,24,27,26,19,25,34)(14,32,36,29,33,35,22,31)

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=ef,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi,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, 377415);