#517959

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

2422 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=f,e^2=f,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, 425155);