#495271

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1292 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224792, #72224798, #72224802, …
maximal normal subgroup of	#72224792, #72224798, #72224802, …
maximal quotient of	C₄×#13261

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402467);