#249217

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1228 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1228 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^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, 156413);