#322425

order	512
nilpotency class	3
isoclinism	isoclinic class of #322409
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,c^2=i,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229621);