#182292

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1344 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226709, #72226711, #72226713
maximal normal subgroup of	#72226709, #72226711, #72226713

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89488);