#2042367

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

1808 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)(3,7,4,6)(9,10)(11,16)(12,15)(13,14)(17,18,21,22)(19,23,20,24);(3,4)(6,7)(9,11,14,12)(10,15,13,16)(18,22)(19,20)(26,29)(30,32);(1,3,5,4)(2,6,8,7)(9,12,14,11)(10,15,13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4,5,3)(2,7,8,6)(17,19,21,20)(18,23,22,24)(25,27,28,31)(26,30,29,32);(1,2)(3,6)(4,7)(5,8)(9,13,14,10)(11,15,12,16)(17,20,21,19)(18,24,22,23)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=fg,d^2=h,e^2=f,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, 1949563);