#146452

order	512
nilpotency class	4
isoclinism	isoclinic class of #13077
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10721
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

863 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,8,13,7,30)(3,14,18,43,17,35,20,51)(4,10,22,37,25,39,24,56)(5,26,27,40,29,57,28,11)(9,31,34,19,33,50,42,16)(15,44,47,64,46,41,48,62)(21,52,53,38,23,55,54,36)(32,59,61,49,60,58,63,45);(1,3,5,19,6,18,27,50,8,17,29,16,7,20,28,31)(2,9,11,35,30,42,57,43,13,33,40,14,12,34,26,51)(4,15,23,45,22,47,54,59,25,46,21,49,24,48,53,58)(10,32,38,44,56,63,52,62,39,60,36,41,37,61,55,64);(1,4)(2,10)(3,15)(5,23)(6,22)(7,24)(8,25)(9,32)(11,38)(12,37)(13,39)(14,41)(16,49)(17,46)(18,47)(19,45)(20,48)(21,29)(26,55)(27,54)(28,53)(30,56)(31,58)(33,60)(34,61)(35,44)(36,40)(42,63)(43,62)(50,59)(51,64)(52,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,h^a=hi,d^b=di,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307184, #72307245, #72307252, …
maximal normal subgroup of	#72307184, #72307245, #72307252, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53648);