#197524

order	512
nilpotency class	3
isoclinism	isoclinic class of #197321
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	C₂³×C₄	C₂⁴
prime radical	#35050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104720);