#353125

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂²×#95
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2358 conjugacy classes of subgroups; only the 1202 normal subgroups are shown.

normal subgroups only

This group has 2358 conjugacy classes of subgroups — too many to draw. Only its 1202 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=i,g^2=i,b^a=be,g^a=gi,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260321);