#517842

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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

2270 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=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, 425038);