#493256

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,11)(8,12)(9,10)(13,14,18,23,19,24,33,39,17,21,30,38,32,16,22,31)(15,25,28,41,29,27,40,43,26,20,34,36,42,37,35,44);(1,2,4,3)(6,10)(11,12)(13,15,17,26)(14,20,21,25)(16,27,24,37)(18,28,30,34)(19,29,32,42)(22,35,33,40)(23,36,38,41)(31,43,39,44);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,21)(15,26)(16,24)(23,38)(28,34)(29,42)(31,39)(35,40);(5,8,9,7)(6,12,10,11)(13,16,30,39,19,14,22,38,17,24,18,31,32,21,33,23)(15,27,34,44,29,20,35,41,26,37,28,43,42,25,40,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eg,d^2=ef,f^2=h,h^2=i,b^a=be,c^a=ci,c^b=ci,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400452);