#519260

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=g,d^2=h,e^2=fg,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, 426456);