#370115

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	#14043
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342076, #72342961, #72342962
maximal normal subgroup of	#72342076, #72342961, #72342962
maximal quotient of	#72343021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277311);