#129482

order	512
nilpotency class	4
isoclinism	isoclinic class of #129435
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2273 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

942 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,f^b=fg,e^c=ei,f^d=fi

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, 36678);