#358818

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#391
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13850
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

953 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)(3,15,19,39)(4,20,24,34)(5,11)(6,12)(7,13)(8,14)(9,18,31,28)(10,23,35,22)(16,32)(17,41,46,59)(21,47,50,57)(25,36)(26,37)(27,38)(29,44)(30,45,54,52)(33,49,58,48)(40,51)(42,55)(43,56)(53,62)(60,63)(61,64);(1,3)(2,9,14,31)(4,16)(5,17)(6,18)(7,28)(8,19)(10,32,35,29)(11,30,38,54)(12,15,13,39)(20,40,34,56)(21,42)(22,43)(23,51)(24,44)(25,45)(26,52)(27,46)(33,55,58,53)(36,41,37,59)(47,60,57,64)(48,61)(49,63)(50,62);(1,4)(2,10,14,35)(3,16)(5,21)(6,22)(7,23)(8,24)(9,29,31,32)(11,33,38,58)(12,34,13,20)(15,40,39,56)(17,42)(18,43)(19,44)(25,48)(26,49)(27,50)(28,51)(30,53,54,55)(36,57,37,47)(41,60,59,64)(45,61)(46,62)(52,63);(1,5,6,25,8,27,7,26)(2,11,12,36,14,38,13,37)(3,17,18,45,19,46,28,52)(4,21,22,48,24,50,23,49)(9,30,15,41,31,54,39,59)(10,33,34,57,35,58,20,47)(16,42,43,61,44,62,51,63)(29,53,40,60,32,55,56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285525, #72285922, #72285960, …
maximal normal subgroup of	#72285525, #72285922, #72285960, …
maximal quotient of	C₂×#358818

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266014);