#200458

order	512
nilpotency class	3
isoclinism	isoclinic class of #200406
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	#35303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1215 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 107654);