#495072

order	512
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#106
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gi,h^b=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285383, #72286646, D₄×#2155, …
maximal normal subgroup of	#72285383, #72286646, D₄×#2155, …
maximal quotient of	#72285455, #72310101

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402268);