#492980

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1082 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)(5,6,10,16,11,17,28,34,12,8,14,25,9,15,26,32)(7,18,22,35,23,13,29,31,20,19,27,36,21,24,30,33);(2,4)(5,7)(6,13,8,24)(9,21)(10,27)(11,23)(12,20)(14,22)(15,18,17,19)(16,33,25,31)(26,29)(28,30)(32,36,34,35);(5,8,10,25,11,15,28,32,12,6,14,16,9,17,26,34)(7,19,22,36,23,24,29,33,20,18,27,35,21,13,30,31);(7,20)(13,24)(18,19)(21,23)(22,27)(29,30)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239417, #72239421, #72239433, …
maximal normal subgroup of	#72239417, #72239421, #72239433, …
maximal quotient of	#72319424, #72320410

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400176);