#423416

order	512
nilpotency class	3
isoclinism	isoclinic class of #423415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15314
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1265 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)(3,9)(4,19,22,38)(5,25)(6,12)(7,29)(8,13)(10,23,37,24)(11,26)(14,35,42,58)(15,44)(16,32)(17,33)(18,34)(20,49,43,30)(21,51,53,36)(27,40)(28,39)(31,46)(41,56,63,64)(45,57)(47,59)(48,54,62,55)(50,60,52,61);(1,3)(2,9)(4,20,22,43)(5,15)(6,17)(7,16)(8,18)(10,35,37,58)(11,31)(12,33)(13,34)(14,24,42,23)(19,30,38,49)(21,52,53,50)(25,44)(26,46)(27,45)(28,47)(29,32)(36,61,51,60)(39,59)(40,57)(41,54,63,55)(48,64,62,56);(1,4,6,24,8,22,7,23)(2,10,12,38,13,37,29,19)(3,14,16,43,18,42,17,20)(5,21,26,55,28,53,27,54)(9,30,32,58,34,49,33,35)(11,36,39,62,40,51,25,48)(15,41,45,52,47,63,46,50)(31,56,44,61,57,64,59,60);(1,5,8,28)(2,11,13,40)(3,15,18,47)(4,21,22,53)(6,26,7,27)(9,31,34,57)(10,36,37,51)(12,39,29,25)(14,41,42,63)(16,45,17,46)(19,48,38,62)(20,50,43,52)(23,54,24,55)(30,56,49,64)(32,44,33,59)(35,60,58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264921, #72264934, #72265047
maximal normal subgroup of	#72264921, #72264934, #72265047
maximal quotient of	#72314025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330612);