#227152

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	#3481	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

609 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,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, 134348);