#343210

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
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×#1624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250406);