#195053

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102249);