#214359

order	512
nilpotency class	3
isoclinism	isoclinic class of #162662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

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, 121555);