#615881

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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

2376 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:

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

Polycyclic presentation:

2^9|c^2=fi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 523077);