#364129

order	512
nilpotency class	3
isoclinism	isoclinic class of #363903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2297
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13960
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1361 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285753, #72285922, #72286143, …
maximal normal subgroup of	#72285753, #72285922, #72286143, …
maximal quotient of	C₂×#364129

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271325);