#855228

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2137 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fghi,c^2=fh,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 762424);