#857918

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

2098 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=gh,e^2=f,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, 765114);