#826411

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gh,c^2=fgh,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 733607);