#343416

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	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 766 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:

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=i,c^2=hi,d^2=h,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, 250612);