#184336

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

723 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,g^2=h,b^a=be,d^a=dhi,e^a=eh,e^b=eh,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91532);