#463392

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	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,d^2=ef,e^2=i,f^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,f^c=fi,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, 370588);