#249336

order	512
nilpotency class	3
isoclinism	isoclinic class of #249319
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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1381 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1381 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,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, 156532);