#6434157

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281914, #72282093
maximal normal subgroup of	#72281914, #72282093

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341353);