#519191

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

2380 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=h,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, 426387);