#113075

order	512
nilpotency class	3
isoclinism	isoclinic class of #113067
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

793 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20271);