#146782

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10788
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

943 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)(3,15)(4,22,25,44)(5,28)(6,11)(7,34)(8,14)(9,17)(10,24,41,59)(12,31)(13,30)(16,50,53,63)(18,37)(19,48)(20,36)(21,49)(23,42,58,43)(26,40,27,46)(29,38)(32,39)(33,45)(35,54,62,55)(47,52,61,57)(51,56,60,64);(1,3,7,19,8,21,30,20)(2,9,13,37,14,39,34,38)(4,16,26,54,25,53,27,55)(5,29,6,32,31,18,33,17)(10,35,42,63,41,62,43,50)(11,36,28,49,45,48,12,15)(22,56,46,47,44,64,40,61)(23,57,59,51,58,52,24,60);(1,4)(2,10)(3,16)(5,24)(6,23)(7,26)(8,25)(9,35)(11,44)(12,46)(13,42)(14,41)(15,47)(17,52)(18,51)(19,54)(20,55)(21,53)(22,45)(27,30)(28,40)(29,60)(31,59)(32,57)(33,58)(34,43)(36,64)(37,63)(38,50)(39,62)(48,56)(49,61)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=eh,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285832, #72307171, #72307179, …
maximal normal subgroup of	#72285832, #72307171, #72307179, …
maximal quotient of	#72285992, #72343249

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53978);