#2044075

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1753 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1951271);