#153725

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
commutator	C₂²	C₂×C₄³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=f,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, 60921);