#263402

order	512
nilpotency class	3
isoclinism	isoclinic class of #263395
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

752 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=dhi,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170598);