#988784

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1780 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=fghi,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 895980);