#194395

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60415	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1285 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101591);