#343413

order	512
nilpotency class	3
isoclinism	isoclinic class of #333407
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,15,10,7,9,13)(8,12,18,20,17,14,16,19)(21,22,23,29,25,28,31,26)(24,32,34,35,33,36,30,27);(1,2)(3,4)(5,7,10,6)(8,14,17,12)(9,13,11,15)(16,19,18,20)(21,23,25,31)(22,26,28,29)(24,30,33,34)(27,32,35,36);(2,4)(5,8,10,17)(6,12,7,14)(9,16,11,18)(13,19,15,20)(21,24,25,33)(22,27,28,35)(23,30,31,34)(26,32,29,36);(5,9)(6,7)(8,16)(10,11)(12,14)(17,18)(22,26)(23,31)(27,32)(28,29)(30,34)(35,36)

Polycyclic presentation:

2^9|a^2=efhi,b^2=hi,c^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250609);