#517718

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

2454 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)(15,20)(16,18)(17,19)(21,22,27,28)(23,24,26,25);(5,7,9,8)(6,10,11,12)(13,15,17,16)(14,18,19,20)(21,23,27,26)(22,25,28,24);(13,14)(15,20)(16,18)(17,19)(21,24,27,25)(22,23,28,26);(1,2,3,4)(5,8,9,7)(6,10,11,12)(13,14)(15,18)(16,20)(17,19)(21,25,27,24)(22,23,28,26);(5,6)(7,10)(8,12)(9,11)(13,16,17,15)(14,18,19,20)(21,26,27,23)(22,25,28,24)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=i,e^2=h,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, 424914);