#903738

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2084 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=i,d^2=f,e^2=fi,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, 810934);