#473493

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380689);