#339758

order	512
nilpotency class	3
isoclinism	isoclinic class of #339703
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 557 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)(3,10)(4,6)(5,9)(7,11)(8,12)(13,14)(15,16)(17,18,22,26)(19,28,20,24)(21,27,23,31)(25,29,32,30);(2,6)(3,11)(4,12)(8,9)(10,14)(13,16)(17,19)(18,24)(20,22)(21,30)(23,29)(25,31)(26,28)(27,32);(1,2)(3,7)(4,8)(5,9)(6,12)(10,11)(13,15)(14,16)(17,20,22,19)(18,25,26,32)(21,30,23,29)(24,27,28,31);(1,3,12,16,5,13,4,11)(2,7,6,14,9,15,8,10)(17,21)(18,26)(19,29)(20,30)(22,23)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 246954);