#495345

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1987
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

803 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307582, #72307787, #72307870, …
maximal normal subgroup of	#72307582, #72307787, #72307870, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402541);