#435119

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

772 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)(5,6,10,17,11,15,33,16)(7,18,21,34,22,12,20,25)(8,23,28,35,26,36,27,13)(9,29,31,24,32,19,30,14);(5,7,10,20,11,22,33,21)(6,12,17,34,15,18,16,25)(8,24,28,29,26,14,27,19)(9,13,30,23,32,35,31,36);(5,8,10,28,11,26,33,27)(6,13,17,23,15,35,16,36)(7,19,20,24,22,29,21,14)(9,25,30,12,32,34,31,18);(2,4)(5,9)(6,14)(7,13)(8,25)(10,30)(11,32)(12,28)(15,24)(16,29)(17,19)(18,27)(20,23)(21,36)(22,35)(26,34)(31,33)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,f^2=i,g^2=i,h^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,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=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, 342315);