#149258

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1999
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 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)(7,21)(8,27)(9,33)(10,19)(11,16)(12,17)(13,20)(14,23)(15,30)(18,36)(22,53)(24,41)(25,48)(26,52)(28,62)(29,49)(31,65)(32,50)(34,45)(35,47)(37,51)(38,46)(39,54)(40,60)(42,59)(43,55)(44,64)(56,68)(57,61)(58,66)(63,67);(5,7,11,25,12,24,36,59,13,26,35,60,10,37,9,23)(6,14,33,51,19,40,47,52,20,42,18,41,17,48,16,21)(8,28,34,55,30,54,64,22,32,58,31,56,38,63,29,57)(15,43,45,62,27,61,49,67,46,68,65,66,50,53,44,39);(5,8,10,30,13,32,12,38)(6,15,17,27,20,46,19,50)(7,22,24,55,26,57,37,56)(9,34,35,64,36,31,11,29)(14,39,40,66,42,67,48,62)(16,45,18,49,47,65,33,44)(21,43,51,53,52,68,41,61)(23,58,25,54,59,28,60,63)

Polycyclic presentation:

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

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, 56454);