#360828

order	512
nilpotency class	3
isoclinism	isoclinic class of #360226
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2352
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13881
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1153 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268024);