#518995

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3714 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 426191);