#212890

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
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	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups; only the 238 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 238 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22,27,32)(23,33,34,29)(24,36,35,28)(25,31,26,30);(5,7,9,16,10,17,8,15)(6,11,12,14,13,19,18,20)(21,23,25,35,27,34,26,24)(22,28,30,33,32,36,31,29);(5,6)(7,11)(8,18)(9,12)(10,13)(14,16)(15,20)(17,19)(21,24,26,34,27,35,25,23)(22,29,31,36,32,33,30,28);(1,3,4,2)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(23,34)(24,35)(28,36)(29,33)

Polycyclic presentation:

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

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, 120086);