#202149

order	512
nilpotency class	3
isoclinism	isoclinic class of #201928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#447	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308593, C₂×#202149
maximal normal subgroup of	#72308593, C₂×#202149
maximal quotient of	C₂×#202149, #72309101, #72309103, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109345);