#193323

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 274 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:

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

Polycyclic presentation:

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

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, 100519);