#1079014

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2094 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 986210);