#535714

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

2442 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)(6,10)(8,12)(13,14)(15,19)(16,17)(18,20)(22,25)(23,27);(5,6,9,10)(7,8,11,12)(14,16)(18,19)(21,22)(23,28)(24,25)(26,27);(6,10)(7,11)(13,15,17,20)(14,18,16,19)(21,22,24,25)(23,26,27,28);(5,7)(6,8)(9,11)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(5,8)(6,7)(9,12)(10,11)(13,16,17,14)(15,19,20,18)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,e^2=fh,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, 442910);