#519665

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=gi,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 426861);