#333203

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

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

normal subgroups only

This group has 691 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:

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

Polycyclic presentation:

2^9|a^2=e,c^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, 240399);