#721617

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2763 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:

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

Polycyclic presentation:

2^9|a^2=fgh,d^2=i,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 628813);