#493791

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1802
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1051 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285693, #72285848, #72285937
maximal normal subgroup of	#72285693, #72285848, #72285937
maximal quotient of	#72321270, #72342589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400987);