#147948

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

958 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4)(5,6)(7,20)(8,25)(9,31)(10,18)(11,16)(12,19)(13,22)(14,28)(15,35)(17,34)(21,42)(23,50)(24,40)(26,60)(27,43)(29,46)(30,45)(32,52)(33,51)(36,49)(37,47)(38,62)(39,61)(41,57)(44,64)(48,63)(53,54)(55,66)(56,67)(58,59)(65,68);(5,7)(6,13)(8,26)(9,32)(10,23)(11,37)(12,24)(14,42)(15,47)(16,39)(17,50)(18,52)(19,41)(20,31)(21,27)(22,35)(25,59)(28,56)(29,65)(30,62)(33,58)(34,57)(36,61)(38,44)(40,49)(43,60)(45,53)(46,66)(48,67)(51,68)(54,63)(55,64);(5,8,12,30)(6,14,19,46)(7,21,24,55)(9,33,36,63)(10,29,11,28)(13,38,41,60)(15,48,17,51)(16,45,18,25)(20,53,40,59)(22,56,57,65)(23,58,37,54)(26,61,62,32)(27,34,64,35)(31,43,49,44)(39,68,52,67)(42,47,66,50)

Polycyclic presentation:

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

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, 55144);