#2044022

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1943 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fh,d^2=gh,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, 1951218);