#121773

order	512
nilpotency class	3
isoclinism	isoclinic class of #112946
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,f^c=fh,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28969);