#185554

order	512
nilpotency class	3
isoclinism	isoclinic class of #185463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3515	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2232 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=ghi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92750);