#212904

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

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

normal subgroups only

This group has 746 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)(5,6,11,16)(7,17,18,13)(8,20,19,12)(9,15,10,14)(23,29)(24,30)(26,32)(28,33)(31,35)(34,36);(5,7,9,19,11,18,10,8)(6,12,14,17,16,20,15,13)(21,22,25,27)(23,26,31,34)(24,28,30,33)(29,32,35,36);(5,8,10,18,11,19,9,7)(6,13,15,20,16,17,14,12)(21,23,30,35,25,31,24,29)(22,26,33,36,27,34,28,32);(1,3,4,2)(7,18)(8,19)(12,20)(13,17)(22,27)(26,34)(28,33)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,c^2=fhi,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, 120100);