#210419

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3531	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=di,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117615);