#197327

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1193 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6,7,8)(9,10)(11,19)(12,23,24,16)(13,26)(14,18)(15,21)(17,28)(20,33,35,40)(22,34)(25,31,27,32)(29,36,39,37)(30,38);(1,2,4,8,5,7,3,6)(9,11,13,21,14,22,28,38)(10,15,17,19,18,30,26,34)(12,20,27,37,24,35,25,36)(16,29,32,33,23,39,31,40);(9,12,14,24)(10,16,18,23)(11,20,22,35)(13,27,28,25)(15,29,30,39)(17,32,26,31)(19,33,34,40)(21,37,38,36);(1,3,5,4)(2,6,7,8)(12,24)(16,23)(20,35)(25,27)(29,39)(31,32)(33,40)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=i,e^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, 104523);