#498068

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₁₆
socle, prime radical	C₂³	C₂³×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^b=de

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, 405264);