#245163

order	512
nilpotency class	3
isoclinism	isoclinic class of #245147
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1201 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1201 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152359);