#457985

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

779 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,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, 365181);