#1506669

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

2002 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=hi,d^2=fgh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1413865);