#899381

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=g,e^2=fghi,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, 806577);