#186951

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	C₂×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3422 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)(9,10,14,20,15,18,13,19)(11,21,25,16,24,31,23,30)(12,17,26,33,27,32,28,34)(22,36,37,40,38,35,39,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,25)(14,23)(15,24)(17,29)(18,30)(19,31)(20,21)(26,39)(27,38)(28,37)(32,40)(33,35)(34,36);(9,12,15,27)(10,17,18,32)(11,22,24,38)(13,28,14,26)(16,29,30,40)(19,34,20,33)(21,35,31,36)(23,39,25,37);(10,18)(16,30)(17,32)(19,20)(21,31)(29,40)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=g,c^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	#72279408, #72281039
maximal normal subgroup of	#72279408, #72281039

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94147);