#492984

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1752
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

997 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,h^2=i,b^a=be,e^a=ei,c^b=ch,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285672, #72285705, #72285725, …
maximal normal subgroup of	#72285672, #72285705, #72285725, …
maximal quotient of	#72310158, #72321442, #72342631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400180);