#1215575

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1784 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)(3,7)(9,10,13,14)(11,12,16,15)(17,18,21,23)(19,20,24,22)(25,26,30,28)(27,32,31,29);(10,14)(12,15)(17,19,21,24)(18,22,23,20)(25,26,30,28)(27,29,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,20,21,22)(18,19,23,24)(25,27)(26,29)(28,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,28,30,26)(27,32,31,29);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,29,30,32)(26,31,28,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgi,c^2=f,d^2=fghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

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, 1122771);