#122043

order	512
nilpotency class	3
isoclinism	isoclinic class of #120247
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1446
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 29239);