#556829

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2582 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464025);