#493109

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

766 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)(7,15)(8,20)(9,14)(10,12)(11,17)(13,21)(16,26)(18,25)(19,24)(22,34)(23,27)(28,30)(29,32)(31,35)(33,36);(2,4)(5,7,8,18,9,19,22,33,10,17,21,32,23,16,30,31)(6,11,13,25,14,26,28,36,12,15,20,29,27,24,34,35);(7,16,17,19)(8,21)(11,24,15,26)(13,20)(18,33,32,31)(22,30)(25,36,29,35)(28,34);(6,12)(11,15)(13,20)(14,27)(24,26)(25,29)(28,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226782, #72226796, #72239416, …
maximal normal subgroup of	#72226782, #72226796, #72239416, …
maximal quotient of	#72285597, #72342629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400305);