#205801

order	512
nilpotency class	3
isoclinism	isoclinic class of #205799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268351, #72268359, #72268373, …
maximal normal subgroup of	#72268351, #72268359, #72268373, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112997);