#495536

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21395
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

808 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=fh,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307828, #72307924, #72307958
maximal normal subgroup of	#72307828, #72307924, #72307958

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402732);