#147355

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
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

705 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265571, #72265577, #72265606
maximal normal subgroup of	#72265571, #72265577, #72265606

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54551);