#358872

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62686	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282010
maximal normal subgroup of	#72282010
maximal quotient of	#72282157

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266068);