#374007

order	512
nilpotency class	3
isoclinism	isoclinic class of #213728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35659	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2459 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

normal subgroups only

This group has 2459 conjugacy classes of subgroups — too many to draw. Only its 346 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,g^c=gi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281203);