#852779

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2507 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=f,d^2=h,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, 759975);