#343114

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
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	#18686	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 679 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:

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

Polycyclic presentation:

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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250310);