#492834

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1726
prime radical	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

841 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239195, #72239196, #72239200
maximal normal subgroup of	#72239195, #72239196, #72239200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400030);