#1079012

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=f,c^2=f,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 986208);