#333189

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#560
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

569 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

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, 240385);