#519103

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

2306 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)(5,6,9,7)(8,11,12,10)(13,14)(15,19)(16,17)(18,20)(22,25)(26,28);(6,7)(10,11)(13,15,17,20)(14,18,16,19);(13,14,17,16)(15,19,20,18)(21,22)(23,26)(24,25)(27,28);(5,7,9,6)(8,10,12,11)(13,15)(14,19)(16,18)(17,20)(21,23,24,27)(22,26,25,28);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,19,20,18)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=fg,e^2=f,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, 426299);