#147239

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10885
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di,e^c=ei

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, 54435);