#520654

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2602 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=i,d^2=fh,e^2=fg,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, 427850);