#471152

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16764
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339028, #72339032, #72339041
maximal normal subgroup of	#72339028, #72339032, #72339041

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378348);