#480128

order	512
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#99
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1459
prime radical	#3449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

967 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,f^e=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387324);