#883879

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

2074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=fgh,d^2=gi,e^2=fgh,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, 791075);