#357805

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2702 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320968, #72320984, #72321006
maximal normal subgroup of	#72320968, #72320984, #72321006
maximal quotient of	#72321769

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265001);