#298923

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1230 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=fh,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206119);