#376768

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14185
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1261 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342961, #72342962, #72342976
maximal normal subgroup of	#72342961, #72342962, #72342976
maximal quotient of	#72343022

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283964);