#110222

order	512
nilpotency class	3
isoclinism	isoclinic class of #110191
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

635 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 635 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,4,8)(3,11,14,15,12,16,13,7)(17,18)(19,28,31,24)(20,32)(21,37)(22,27)(23,26)(25,35)(29,44,33,46)(30,40,38,41)(34,47)(36,43)(39,45,42,48);(1,3)(2,7,10,15)(4,13)(5,14)(6,12)(8,16,9,11)(17,19,22,30,23,31,21,38)(18,24,27,41,26,28,37,40)(20,33,36,48,34,29,35,45)(25,42,32,46,43,39,47,44);(2,8,10,9)(3,12)(7,11,15,16)(13,14)(17,20,21,35,23,34,22,36)(18,25,27,32,26,43,37,47)(19,29,38,48,31,33,30,45)(24,39,41,44,28,42,40,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,c^b=ch,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257501, #72286487, #72286739, …
maximal normal subgroup of	#72257501, #72286487, #72286739, …
maximal quotient of	#72257737

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17418);