#433977

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

984 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)(7,19)(8,14,27,9)(10,17)(11,18)(12,16)(13,22)(15,30,26,29)(20,28,35,34)(21,36,31,25)(23,32)(24,33);(5,7)(6,13)(8,25)(9,28)(10,23)(11,24)(12,22)(14,34)(15,35)(16,33)(17,19)(18,32)(20,26)(21,29)(27,36)(30,31);(1,2,3,4)(5,8,12,15,11,27,10,26)(6,14,16,30,18,9,17,29)(7,20,22,25,24,35,23,36)(13,31,33,28,32,21,19,34);(5,9)(6,15)(7,21)(8,17)(10,30)(11,14)(12,29)(13,25)(16,27)(18,26)(19,20)(22,34)(23,28)(24,31)(32,36)(33,35)

Polycyclic presentation:

2^9|a^2=eg,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, 341173);