#542513

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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

3234 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)(6,7)(13,14,18,19)(15,20,23,27)(16,21,22,17)(24,25,26,28);(1,2,4,3)(9,10,11,12)(14,19)(15,23)(17,21)(24,26);(5,6,8,7)(9,10)(11,12)(13,15)(14,20)(16,26)(17,25)(18,23)(19,27)(21,28)(22,24);(5,7,8,6)(13,16,18,22)(14,21,19,17)(15,24,23,26)(20,25,27,28);(1,3,4,2)(13,17,18,21)(14,22,19,16)(15,25,23,28)(20,24,27,26)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 449709);