#214083

order	512
nilpotency class	3
isoclinism	isoclinic class of #195972
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1215 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)(25,26,29,33,30,34,28,32)(27,31,36,39,35,38,37,40);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(27,35)(31,38)(36,37)(39,40);(3,8)(5,7)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,27)(26,31)(28,37)(29,36)(30,35)(32,40)(33,39)(34,38);(2,6)(5,7)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,28,30,29)(26,32,34,33)(27,36,35,37)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^c=di,g^c=gh

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, 121279);