#333158

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	#18686	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 641 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=eh,b^2=i,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, 240354);