#186953

order	512
nilpotency class	3
isoclinism	isoclinic class of #186835
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62811	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3414 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)(9,10,14,18,15,20,13,19)(11,21,23,16,25,31,24,30)(12,17,27,32,28,33,26,34)(22,36,37,40,38,35,39,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,23)(14,24)(15,25)(17,29)(18,21)(19,31)(20,30)(26,37)(27,39)(28,38)(32,35)(33,40)(34,36);(9,12)(10,17)(11,22)(13,26)(14,27)(15,28)(16,29)(18,32)(19,34)(20,33)(21,35)(23,37)(24,39)(25,38)(30,40)(31,36);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,26,28,27)(16,21,30,31)(17,32,33,34)(22,37,38,39)(29,35,40,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281039
maximal normal subgroup of	#72281039

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94149);