#468014

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 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)(3,4)(5,6,9,23,10,14,24,13)(7,15,18,11,17,31,35,26)(8,19,20,28,22,12,21,29)(16,32,36,27,34,30,33,25);(5,7)(6,11,14,26)(8,16)(9,18)(10,17)(12,27,19,25)(13,15,23,31)(20,36)(21,33)(22,34)(24,35)(28,32,29,30);(5,8,9,20,10,22,24,21)(6,12,13,28,14,19,23,29)(7,16,18,36,17,34,35,33)(11,25,15,30,26,27,31,32);(2,4)(7,17)(8,20,22,21)(11,26)(12,28,19,29)(15,31)(16,33,34,36)(18,35)(25,32,27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280002
maximal normal subgroup of	#72280002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375210);