#519617

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

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

3378 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,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, 426813);