#471502

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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)(5,6,10,14,11,15,28,21)(7,16,19,31,20,32,26,12)(8,9,22,24,23,25,13,27)(17,33,30,29,34,36,35,18);(5,7,10,26,11,20,28,19)(6,12,14,32,15,31,21,16)(8,17,13,30,23,34,22,35)(9,18,27,33,25,29,24,36);(2,4)(5,8)(6,9)(7,17)(10,13)(11,23)(12,29)(14,27)(15,25)(16,33)(18,31)(19,35)(20,34)(21,24)(22,28)(26,30)(32,36);(5,9)(6,13)(7,18)(8,21)(10,27)(11,25)(12,30)(14,23)(15,22)(16,17)(19,36)(20,29)(24,28)(26,33)(31,35)(32,34)

Polycyclic presentation:

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

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, 378698);