#535382

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5134 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 442578);