#516554

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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

2972 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)(6,8)(10,12)(13,14,18,16)(15,20,17,19)(22,26)(24,28);(5,6,9,8)(7,10,11,12)(13,15,18,17)(14,19,16,20)(21,22)(23,24)(25,26)(27,28);(7,11)(10,12)(14,16)(19,20)(22,26)(23,27);(5,7)(6,10)(8,12)(9,11)(13,16,18,14)(15,20,17,19)(21,23)(22,24)(25,27)(26,28);(5,8,9,6)(7,12,11,10)(13,17,18,15)(14,20,16,19)(21,24)(22,23)(25,28)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423750);