#210423

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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

728 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:

(1,2,7,12,8,13,6,27)(3,14,16,9,18,20,17,28)(4,19,22,10,23,15,21,29)(5,11,25,30,26,31,24,32)(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)(4,11)(5,15)(6,17)(7,16)(8,18)(10,24)(12,20)(13,28)(14,27)(19,26)(21,32)(22,30)(23,31)(25,29)(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)(3,11)(5,20)(6,21)(7,22)(8,23)(9,24)(12,15)(13,29)(14,26)(16,30)(17,32)(18,31)(19,27)(25,28)(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);(1,5,7,25,8,26,6,24)(2,11,12,30,13,31,27,32)(3,15,16,29,18,19,17,10)(4,20,22,28,23,14,21,9)(35,44)(36,47)(39,54)(40,56)(45,50)(46,58)(48,51)(55,57)

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=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 117619);