#148387

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dhi,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307228, #72307236, #72307258, …
maximal normal subgroup of	#72307228, #72307236, #72307258, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55583);