#918771

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2951 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fgh,d^2=fg,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, 825967);