#206258

order	512
nilpotency class	3
isoclinism	isoclinic class of #19697
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1274 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,29)(24,38)(25,34)(26,36)(28,37)(30,41)(31,35)(33,39)(40,43)(42,44);(2,4)(5,7,11,6,9,12,8,10)(13,15,19,26)(14,20,23,36)(16,24,30,40)(17,31,18,25)(21,33,37,44)(22,35,32,34)(27,38,41,43)(28,42,29,39);(6,10)(7,12)(13,16,17,28,19,30,18,29)(14,21,32,41,23,37,22,27)(15,24,31,42,26,40,25,39)(20,33,34,43,36,44,35,38);(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,28,30,29)(20,34,36,35)(21,27,37,41)(24,39,40,42)(33,43,44,38)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fhi,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113454);