#249265

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
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×#1598	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1346 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1346 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156461);