#249229

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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1295 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1295 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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, 156425);