#348828

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×D₄²	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1153 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1153 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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, 256024);