#182045

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1266 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89241);