#519180

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

2922 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)(7,11)(8,9)(10,12)(13,14,16,19)(15,18,20,17)(21,22,25,27)(23,28,24,26);(6,9)(10,11)(13,14)(15,17)(16,19)(18,20)(21,23)(22,26)(24,25)(27,28);(13,15)(14,17)(16,20)(18,19)(21,22)(23,28)(24,26)(25,27);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,23)(22,26,27,28);(5,7,8,12)(6,10,9,11)(13,14)(15,17)(16,19)(18,20)(21,22)(23,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=i,d^2=h,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, 426376);