#199079

order	512
nilpotency class	3
isoclinism	isoclinic class of #198798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=hi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239440, #72280154, #72280206, …
maximal normal subgroup of	#72239440, #72280154, #72280206, …
maximal quotient of	#72257673

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 106275);