#212748

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=eh,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119944);