#482053

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1114 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)(3,5)(4,7)(6,8)(9,10)(11,20,23,30)(12,17,14,28)(13,18)(15,19)(16,21)(22,36,25,39)(24,33,26,40)(27,35)(29,32)(31,37)(34,38);(2,5)(3,8)(6,7)(9,11)(10,16)(12,21)(13,25)(14,29)(15,23)(17,30)(18,34)(19,32)(20,28)(22,27)(24,38)(26,37)(31,35)(33,36)(39,40);(9,12,15,14)(10,17,19,28)(11,21,23,29)(13,24,27,26)(16,30,32,20)(18,33,35,40)(22,37,25,38)(31,39,34,36);(9,13,15,27)(10,18,19,35)(11,22,23,25)(12,24,14,26)(16,31,32,34)(17,33,28,40)(20,36,30,39)(21,37,29,38)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389249);