#577914

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2914 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)(7,11)(9,12)(14,17)(15,19)(21,22,25,26)(23,27,28,24);(5,6,8,10)(7,9,11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,24)(25,28)(26,27);(5,7)(6,9)(8,11)(10,12)(13,15,16,19)(14,18,17,20)(21,24,25,27)(22,23,26,28);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,24)(25,26)(27,28);(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,23)(25,27)(26,28)

Polycyclic presentation:

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

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, 485110);