#333264

order	512
nilpotency class	3
isoclinism	isoclinic class of #333199
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 581 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:

(1,2,6,11,7,3,5,9)(4,8,14,12,15,10,13,16)(17,18,23,28)(19,24,21,31)(20,30,25,29)(22,27,32,26);(1,3,7,2)(4,12,15,16)(5,9,6,11)(8,13,10,14)(17,19)(18,24)(20,27)(21,23)(22,29)(25,26)(28,31)(30,32);(1,3)(2,7)(4,10)(5,11)(6,9)(8,15)(12,13)(14,16)(17,20,23,25)(18,21,28,19)(22,31,32,24)(26,29,27,30);(1,4,7,15)(2,8,3,10)(5,14,6,13)(9,12,11,16)(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=hi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=defh,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240460);