#249038

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
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	#36961	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1353 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1353 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156234);