#504179

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343620, #72343829, #72343857
maximal normal subgroup of	#72343620, #72343829, #72343857

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411375);