#343095

order	512
nilpotency class	3
isoclinism	isoclinic class of #343075
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	#3169	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=hi,c^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defhi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250291);