#1687402

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2172 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=ghi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1594598);