#188009

order	512
nilpotency class	3
isoclinism	isoclinic class of #187928
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62811	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3416 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)(3,7,8,5)(4,6)(9,10,14,19,15,20,13,18)(11,16,24,31,23,30,22,29)(12,17,27,34,26,33,25,32)(21,35,38,39,37,28,36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,22)(14,24)(15,23)(17,28)(18,29)(19,31)(20,30)(25,36)(26,37)(27,38)(32,39)(33,35)(34,40);(9,12)(10,17)(11,21)(13,25)(14,27)(15,26)(16,28)(18,32)(19,34)(20,33)(22,36)(23,37)(24,38)(29,39)(30,35)(31,40);(2,6)(5,7)(9,13,15,14)(10,18,20,19)(11,22,23,24)(12,25,26,27)(16,29,30,31)(17,32,33,34)(21,36,37,38)(28,39,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95205);