#249443

order	512
nilpotency class	3
isoclinism	isoclinic class of #249427
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	#36721	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1321 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1321 conjugacy classes of subgroups — too many to draw. Only its 148 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18,21,23)(19,25,26,32)(20,28,30,24)(22,29,31,27);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(22,31)(25,32);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,19)(18,22)(20,29)(21,26)(23,31)(24,32)(25,28)(27,30);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(18,23)(19,26)(20,30)(25,32)

Polycyclic presentation:

2^9|a^2=h,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156639);