#108589

order	512
nilpotency class	3
isoclinism	isoclinic class of #8610
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

448 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318993, #72343141
maximal normal subgroup of	#72318993, #72343141
maximal quotient of	#72319246

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15785);