#214107

order	512
nilpotency class	3
isoclinism	isoclinic class of #195981
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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

947 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)(9,10,13,17,14,18,12,16)(11,15,20,23,19,22,21,24)(25,26,28,33,29,32,37,40)(27,30,35,38,34,31,36,39);(1,3)(2,5)(4,8)(6,7)(11,19)(15,22)(20,21)(23,24)(25,27,29,34)(26,30,32,31)(28,35,37,36)(33,38,40,39);(3,8)(5,7)(9,11)(10,15)(12,21)(13,20)(14,19)(16,24)(17,23)(18,22)(25,27)(26,31)(28,36)(29,34)(30,32)(33,38)(35,37)(39,40);(2,6)(5,7)(9,12,14,13)(10,16,18,17)(11,20,19,21)(15,23,22,24)(27,34)(30,31)(35,36)(38,39)

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^b=dh,d^c=di,g^c=gh

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, 121303);