#469927

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16669
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

647 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)(3,8)(4,16,18,31)(5,21)(6,10)(7,11)(9,20,26,19)(12,27)(13,29,17,32)(14,24)(15,22)(23,30,25,28);(3,12,14,15)(8,22,24,27)(13,28,17,30)(23,29,25,32);(1,3)(2,8)(4,17,18,13)(5,15)(6,12)(7,14)(9,25,26,23)(10,27)(11,24)(16,29,31,32)(19,28,20,30)(21,22);(1,4,5,19,7,18,6,20)(2,9,10,16,11,26,21,31)(3,13,15,30,14,17,12,28)(8,23,27,32,24,25,22,29)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319564
maximal normal subgroup of	#72319564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377123);