#519537

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

2780 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fghi,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, 426733);