#123582

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237699, #72237702, #72237708, …
maximal normal subgroup of	#72237699, #72237702, #72237708, …
maximal quotient of	#72304148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30778);