#451172

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15987
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

745 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)(3,14)(4,20)(5,25,28,11)(6,29)(7,12)(8,13)(9,17)(10,23)(15,35)(16,33,26,39)(18,34)(19,43)(21,42)(22,49,54,63)(24,37)(27,40,31,41)(30,38)(32,47)(36,52,55,53)(44,61,56,62)(45,58)(46,60,48,59)(50,64,51,57);(1,3,6,18,8,17,7,19)(2,9,12,34,13,14,29,43)(4,21,23,45,24,47,30,15)(5,26,27,48,28,16,31,46)(10,35,20,32,38,58,37,42)(11,39,40,60,25,33,41,59)(22,51,52,56,54,50,53,44)(36,62,63,57,55,61,49,64);(1,4)(2,10)(3,15,17,45)(5,22)(6,30)(7,23)(8,24)(9,32,14,42)(11,36)(12,37)(13,38)(16,44,26,56)(18,47,19,21)(20,29)(25,55)(27,53)(28,54)(31,52)(33,57,39,64)(34,35,43,58)(40,49)(41,63)(46,50,48,51)(59,62,60,61);(1,5)(2,11)(3,16)(4,22)(6,31)(7,27)(8,28)(9,33)(10,36)(12,41)(13,25)(14,39)(15,44)(17,26)(18,48)(19,46)(20,49)(21,50)(23,53)(24,54)(29,40)(30,52)(32,57)(34,60)(35,61)(37,63)(38,55)(42,64)(43,59)(45,56)(47,51)(58,62)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358368);