#518009

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3082 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gi,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, 425205);