#110893

order	512
nilpotency class	3
isoclinism	isoclinic class of #110892
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

581 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^c=eh,f^c=fh

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, 18089);