#365128

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	#13971
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342962, #72342963, #72342976
maximal normal subgroup of	#72342962, #72342963, #72342976
maximal quotient of	#72343055

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272324);