#343024

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	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 726 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,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, 250220);