#554325

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 461521);