#918024

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3742 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 825220);