#348080

order	512
nilpotency class	3
isoclinism	isoclinic class of #348070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3483	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=gi,f^d=fi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255276);