#474042

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
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

759 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=egh,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,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, 381238);