#182917

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90113);