#2826570

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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

1957 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2733766);