#348832

order	512
nilpotency class	3
isoclinism	isoclinic class of #346740
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 873 conjugacy classes of subgroups — too many to draw. Only its 176 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=h,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=dh,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 256028);