#2044045

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

1927 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=gh,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, 1951241);