#519093

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×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

4922 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,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, 426289);