#517402

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×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

2746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,3)(5,6,7,8)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32);(1,2,4,3)(10,13)(14,16)(18,23)(19,25)(24,31)(26,27);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,32)(26,29)(28,31);(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,21)(18,19)(20,28)(22,30)(23,25)(24,26)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,d^2=f,e^2=ghi,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, 424598);