#121142

order	512
nilpotency class	3
isoclinism	isoclinic class of #121137
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

460 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:

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

Polycyclic presentation:

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

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, 28338);