#845284

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2106 conjugacy classes of subgroups; only the 654 normal subgroups are shown.

normal subgroups only

This group has 2106 conjugacy classes of subgroups — too many to draw. Only its 654 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752480);