#129486

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2273 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 674 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) = 64

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

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

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,e^b=ei,f^b=fg,d^c=di,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, 36682);