#182862

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1270 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)(3,7)(4,6)(5,8)(9,10,14,18,15,19,13,28)(11,20,24,30,22,34,23,16)(12,17,27,32,25,31,26,38)(21,33,36,39,37,40,35,29);(1,3)(2,5,6,7)(4,8)(9,11)(10,16,19,30)(12,21)(13,23)(14,24)(15,22)(17,29,31,39)(18,20,28,34)(25,37)(26,35)(27,36)(32,33,38,40);(9,12,14,27,15,25,13,26)(10,17,18,32,19,31,28,38)(11,21,24,36,22,37,23,35)(16,29,20,33,30,39,34,40);(11,22)(12,25)(16,30)(17,31)(20,34)(23,24)(26,27)(32,38)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90058);