#333134

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

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

normal subgroups only

This group has 683 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,c^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, 240330);