#249382

order	512
nilpotency class	3
isoclinism	isoclinic class of #249355
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	#35356	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1335 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1335 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156578);