#195800

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1392 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,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, 102996);