#532544

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2582 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=ghi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=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, 439740);