#149292

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(5,6)(7,12)(8,10)(9,11)(13,14,18,24,17,23,32,41,19,25,34,42,16,22,31,40)(15,26,29,43,27,21,38,39,30,36,33,44,28,20,35,37);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,28)(17,27)(18,33)(19,30)(21,25)(22,36)(23,26)(24,39)(29,34)(31,38)(32,35)(37,42)(40,43)(41,44);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(14,21)(16,28)(17,27)(18,33)(19,30)(20,25)(22,26)(23,36)(24,37)(29,34)(31,38)(32,35)(39,42)(40,44)(41,43)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,e^b=ei,g^b=gi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56488);