#517974

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

2086 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)(13,14,17,16)(15,18,19,20)(23,27)(26,28);(5,7,10,8)(6,11,9,12)(15,19)(18,20)(21,22,25,24)(23,28,27,26);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,15)(14,18)(16,20)(17,19)(21,23,25,27)(22,26,24,28);(1,3,4,2)(5,8,10,7)(6,11,9,12)(13,16,17,14)(15,20,19,18)(21,24,25,22)(23,28,27,26);(5,9,10,6)(7,12,8,11)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=fghi,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, 425170);