#557282

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2030 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=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, 464478);