#352073

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

745 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 745 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259269);