#343026

order	512
nilpotency class	3
isoclinism	isoclinic class of #343003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250222);