#532533

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

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

2534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 439729);