#493325

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

630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dei,c^b=ce

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, 400521);