#188021

order	512
nilpotency class	3
isoclinism	isoclinic class of #187944
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

3402 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)(9,10,14,18,15,19,13,27)(11,16,23,29,22,30,21,37)(12,17,26,31,25,32,24,38)(20,33,36,40,35,28,34,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,28)(18,29)(19,30)(24,34)(25,35)(26,36)(27,37)(31,39)(32,33)(38,40);(9,12)(10,17)(11,20)(13,24)(14,26)(15,25)(16,28)(18,31)(19,32)(21,34)(22,35)(23,36)(27,38)(29,39)(30,33)(37,40);(9,13,15,14)(10,18,19,27)(11,21,22,23)(12,24,25,26)(16,29,30,37)(17,31,32,38)(20,34,35,36)(28,39,33,40)

Polycyclic presentation:

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

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, 95217);