#585538

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

2620 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 492734);