#148330

order	512
nilpotency class	4
isoclinism	isoclinic class of #13312
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

658 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)(7,19)(8,25)(9,29)(10,18)(11,16)(12,17)(13,22)(14,27)(15,20)(21,49)(23,55)(24,46)(26,58)(28,43)(30,41)(31,40)(32,45)(33,39)(34,62)(35,60)(36,54)(37,65)(38,59)(42,47)(44,51)(48,57)(50,56)(52,66)(53,67)(61,63)(64,68);(5,7,20,47,11,23,31,54,12,24,30,60,10,32,9,22)(6,13,29,45,18,35,41,46,17,36,40,55,16,42,15,19)(8,21,48,65,27,52,59,68,28,53,61,62,33,51,26,50)(14,37,57,49,25,56,58,44,39,34,63,67,43,64,38,66);(5,8)(6,14)(7,21)(9,26)(10,33)(11,27)(12,28)(13,34)(15,38)(16,43)(17,39)(18,25)(19,44)(20,48)(22,50)(23,52)(24,53)(29,57)(30,61)(31,59)(32,51)(35,64)(36,37)(40,63)(41,58)(42,56)(45,67)(46,66)(47,65)(49,55)(54,68)(60,62)

Polycyclic presentation:

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

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, 55526);