#285442

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,g^b=ghi,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192638);