#343127

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	#18696	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 641 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=efhi,d^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, 250323);