#147479

order	512
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10931
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314212
maximal normal subgroup of	#72314212

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54675);