#450227

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1065 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)(7,19)(8,9,26,14)(10,18)(11,16)(12,17)(13,22)(15,29,30,28)(20,35,36,27)(21,25,31,32)(23,33)(24,34);(1,2)(3,4)(5,7)(6,13)(8,25)(9,27)(10,24)(11,22)(12,23)(14,35)(15,20)(16,33)(17,34)(18,19)(21,28)(26,32)(29,31)(30,36);(2,4)(5,8,10,15,12,26,11,30)(6,14,18,28,17,9,16,29)(7,20,24,32,23,36,22,25)(13,31,19,35,34,21,33,27);(5,9)(6,15)(7,21)(8,16)(10,29)(11,28)(12,14)(13,32)(17,30)(18,26)(19,36)(20,33)(22,35)(23,31)(24,27)(25,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357423);