#249462

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	#36704	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1400 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1400 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)(4,8)(5,13)(6,9)(10,16)(11,14)(12,15)(17,18)(19,25)(20,28)(21,23)(22,27)(24,30)(26,32)(29,31);(1,3,6,5)(2,7,9,13)(4,10,12,11)(8,14,15,16)(22,31)(25,32);(1,4)(2,8)(3,10)(5,11)(6,12)(7,14)(9,15)(13,16)(17,19,21,26)(18,22,23,31)(20,29,30,27)(24,32,28,25);(1,5,6,3)(2,7,9,13)(4,11,12,10)(8,14,15,16)(18,23)(19,26)(20,30)(25,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=i,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, 156658);