#519397

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

2922 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)(9,10,13,11)(12,15,16,14)(17,18,21,22)(19,23,24,20);(5,6,7,8)(10,11)(14,15)(18,22)(19,24);(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,19)(18,20)(21,24)(22,23);(1,3,4,2)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,22)(21,24);(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,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, 426593);