#249224

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1147 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1147 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156420);