#517785

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=i,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424981);