#918773

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2788 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)(19,20)(21,29)(22,23)(24,30)(25,28)(26,32)(27,31);(6,8)(9,10,12,14)(11,15,16,13)(18,23)(19,25)(24,29)(26,31);(5,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,25)(18,20,23,28)(21,26,30,31)(24,27,29,32);(1,2)(3,4)(9,11,12,16)(10,13,14,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,31)(30,32);(17,21,22,30)(18,24,23,29)(19,26,25,31)(20,27,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=f,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, 825969);