#518212

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2198 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,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, 425408);