#213304

order	512
nilpotency class	3
isoclinism	isoclinic class of #19671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	C₂³	#532
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fi,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120500);