#231564

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

650 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 226 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=fi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138760);