#611094

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#16291	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3314 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 518290);