#227149

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#3483	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=deh,e^b=eh,d^c=dhi,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134345);