#520455

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
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

2234 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=i,d^2=fghi,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427651);