#517752

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

2278 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)(7,12)(8,10)(9,11)(13,14,16,18)(15,17,19,20)(21,22,25,24)(23,27,28,26);(5,7,9,8)(6,10,11,12)(23,28)(26,27);(1,2,3,4)(13,14)(15,17)(16,18)(19,20)(21,23,25,28)(22,26,24,27);(5,6)(7,10)(8,12)(9,11)(21,22,25,24)(23,26,28,27);(5,8,9,7)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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, 424948);