#512646

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
socle	C₂	C₂×#2127
prime radical	#60797	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

810 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)(3,15,18,44)(4,10)(5,11)(6,26)(7,13,30,39)(8,14)(9,17,33,50)(12,29,41,25)(16,43,48,62)(19,32,20,45)(21,52,49,57)(22,35)(23,37,54,61)(24,38)(27,42,51,34)(28,55)(31,46,58,63)(36,53,40,56)(47,60,64,59);(1,3)(2,9)(4,21)(5,17)(6,27)(7,18)(8,20)(10,31)(11,32)(12,15)(13,42)(14,34)(16,24)(19,30)(22,47)(23,49)(25,50)(26,44)(28,46)(29,51)(33,39)(35,57)(36,52)(37,60)(38,59)(40,43)(41,45)(48,54)(53,64)(55,62)(56,63)(58,61);(1,4)(2,10)(3,16)(5,22)(6,28)(7,23)(8,24)(9,31)(11,35)(12,40)(13,37)(14,38)(15,43)(17,46)(18,48)(19,49)(20,21)(25,53)(26,55)(27,47)(29,56)(30,54)(32,57)(33,58)(34,59)(36,41)(39,61)(42,60)(44,62)(45,52)(50,63)(51,64);(1,5,8,6)(2,11,14,26)(3,17,20,27)(4,22,24,28)(7,25,30,29)(9,32,34,44)(10,35,38,55)(12,39,41,13)(15,33,45,42)(16,46,21,47)(18,50,19,51)(23,53,54,56)(31,57,59,62)(36,37,40,61)(43,58,52,60)(48,63,49,64)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=eh,f^b=fghi,g^b=gi,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284730
maximal normal subgroup of	#72284730

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419842);