#517841

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=g,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, 425037);