#147356

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10921
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

703 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265571, #72265582, #72265606
maximal normal subgroup of	#72265571, #72265582, #72265606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54552);