#105616

order	512
nilpotency class	3
isoclinism	isoclinic class of #105616
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,f^b=fi,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233122, #72233128, #72233130
maximal normal subgroup of	#72233122, #72233128, #72233130

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12812);