#147130

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1985
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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)(8,13)(9,14)(10,12)(16,24,26,19)(18,29,32,25)(20,22)(21,28)(23,27)(30,35)(31,36,33,34);(5,7,8,18,9,19,21,34,10,17,22,32,23,24,35,36)(6,11,13,25,14,26,28,33,12,15,20,29,27,16,30,31);(2,4)(5,6)(7,16,17,26)(8,20)(9,14)(10,12)(11,24,15,19)(13,22)(18,33,32,31)(21,30)(23,27)(25,34,29,36)(28,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239196, #72239201, #72239245, …
maximal normal subgroup of	#72239196, #72239201, #72239245, …
maximal quotient of	#72239316

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54326);