#988200

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fghi,d^2=g,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, 895396);