#519116

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

2354 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)(5,6,9,10)(7,11,12,8)(14,19)(17,20)(22,24)(26,27);(6,10)(7,12)(13,14)(15,20)(16,17)(18,19)(21,22,25,24)(23,27,28,26);(5,7)(6,8)(9,12)(10,11)(13,15,18,16)(14,20,19,17)(21,22,25,24)(23,26,28,27);(5,7)(6,11)(8,10)(9,12)(13,16,18,15)(14,20,19,17)(21,23)(22,26)(24,27)(25,28);(5,8,9,11)(6,12,10,7)(13,17,18,20)(14,16,19,15)(21,24,25,22)(23,26,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fh,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, 426312);