#392905

order	512
nilpotency class	3
isoclinism	isoclinic class of #32947
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2847
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14697
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1351 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)(3,9,18,30,17,31,16,45)(4,19,22,33,23,47,21,10)(5,11,26,35,24,34,25,51)(14,36,40,53,39,57,38,28)(15,41,44,63,42,29,43,54)(20,46,50,55,48,64,49,32)(37,58,61,52,59,56,60,62);(1,3,7,18,8,17,6,16)(2,9,12,30,13,31,27,45)(4,14,22,40,23,39,21,38)(5,15,26,44,24,42,25,43)(10,28,19,36,33,53,47,57)(11,29,35,54,34,41,51,63)(20,37,50,61,48,59,49,60)(32,52,46,56,55,62,64,58);(1,4,6,21,8,23,7,22)(2,10,12,19,13,33,27,47)(3,14,16,38,17,39,18,40)(5,20,25,49,24,48,26,50)(9,28,30,36,31,53,45,57)(11,32,35,46,34,55,51,64)(15,37,43,60,42,59,44,61)(29,52,54,56,41,62,63,58);(1,5)(2,11)(3,15)(4,20)(6,25)(7,26)(8,24)(9,29)(10,32)(12,35)(13,34)(14,37)(16,43)(17,42)(18,44)(19,46)(21,49)(22,50)(23,48)(27,51)(28,52)(30,54)(31,41)(33,55)(36,56)(38,60)(39,59)(40,61)(45,63)(47,64)(53,62)(57,58)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281697, #72281785, #72281842, …
maximal normal subgroup of	#72281697, #72281785, #72281842, …
maximal quotient of	#72343149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300101);