#460609

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

977 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^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, 367805);