#122138

order	512
nilpotency class	3
isoclinism	isoclinic class of #122137
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

602 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi,e^c=eh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29334);