#617365

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#525	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6,7,8)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26);(9,10,11,12)(14,18)(15,21)(20,23)(22,27);(5,6)(7,8)(9,10)(11,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(1,2)(3,4)(5,6)(7,8)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 524561);