#495799

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3050	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224944, #72224962, #72224965
maximal normal subgroup of	#72224944, #72224962, #72224965
maximal quotient of	#72323193

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402995);