#105845

order	512
nilpotency class	3
isoclinism	isoclinic class of #105838
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gi,d^b=di,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316882, #72316961
maximal normal subgroup of	#72316882, #72316961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13041);