#472032

order	512
nilpotency class	3
isoclinism	isoclinic class of #464193
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2657
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16810
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1347 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)(3,9,5,20)(4,10)(6,13)(7,22)(8,12)(11,15,30,16)(14,31,21,24)(17,25,32,26)(18,29)(19,28)(23,27);(1,3,6,5)(2,9,13,20)(4,17)(7,15,8,16)(10,25)(11,22,30,12)(14,23)(18,32)(19,21)(24,28)(26,29)(27,31);(1,4)(2,10)(3,14)(5,21)(6,18)(7,23)(8,19)(9,24)(11,26)(12,28)(13,29)(15,32)(16,17)(20,31)(22,27)(25,30);(1,5,6,3)(2,11)(4,14,18,21)(7,16,8,15)(9,22)(10,26)(12,20)(13,30)(17,23,32,19)(24,27)(25,29)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284607, #72287274, #72287345, …
maximal normal subgroup of	#72284607, #72287274, #72287345, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379228);