#517126

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 424322);